I'm not a good spinner. But I've spent an unbelievable amount of time trying, under the direction of many coaches. And I've tried a huge number of different approaches.
I've owned many types of skate or dance spinner, as well as cheaper but more stable exercise related "spin trainers", and "twist trainers" (most of which can also spin), which worked better for me. The latter are often available in the $10-$15 (US dollar) range, though be sure to check on maximum weight load. I like best the ones on which the top surface isn't completely smooth, but as a texture to hold your foot in place better. Five Below used to sell a $5 model, but it's gone for now.
If you go to a gym, they may provide some types of spin training devices for no extra charge. PTs often have them too.
I also tried spin/twist trainers that had attached ropes or stretch cords. They helped me stay balanced on the device, but they may make balance
too easy to maintain. You can usually just leave the ropes or cords on the floor if you don't want to use them. (I've seen some people practice ice on spins with similar ropes or cords, tied down to near the boots. I don't know if it helps.)
None of them help me spin on ice as much as I hoped, partly because I think my reflexes aren't fast enough, and I don't have as much body awareness of my exact orientation, and partly because spinning on a bar or plate or similar device is substantially different from so-called spinning on ice - both because of the elevated skate boot heel (which I suspect is my biggest problem, because of my flexibility limits, and one of the reasons I want a lower heel), and because carving a small circle on ice, balanced off center on edge near the ball of your foot, is different from truly spinning around the center of the foot, in place.
E.g., on a skate spin, gliding around that circle creates centrifugal force outwards, so you may need to align your vertical axis inwards towards the center of the circle. You can get a little of the same effect off ice by placing your foot offset to the side (and perhaps forward or back, to emulate the effects of the high heel) on the spinner - but for that you need one that is pretty stable.
The bar type abrades the floor, so I eventually to switched to spinning on slippery thin synthetic socks, on a polished hardwood floor, or on waxed plastic kitchen floors, if they aren't tiled. That may actually work best - except it doesn't really force you to balance, and you can't spin as fast as on ball bearings. Cotton and wool isn't very slippery, and work poorly.
The others, that have ball bearings, made for interesting pre-skate warm-up exercises, and other people found them interesting to try too - but beware that falling is likely, and you need fall differently on the high friction rubber matting of an ice rink than on low friction ice. As with ice falls, it helps to wear long sleeve clothing, and something over your legs, so you don't get abrasions, and find ways to spread out the impact on as much of your body as possible. It's up to you whether you let other interested parties try try using your equipment - I suppose someone might hurt themselves and sue you.
There is a fun choreographed dance fall you might play with too - you start on a spin, and kind of spiral down to a sitting position on the floor. It works both on and off ice. Maybe not so useful for skating per se, but it's a super gentle fall if you have to fall out of a spin. (Alas, for ordinary non-spun skating, you often can't start spinning fast enough mid-fall to make it work.)
The most expensive two I tried, designed for skaters, had very low quality erratic high friction ball bearings - they were not well made, so they spun in spurts, which really didn't help. And they had the greatest friction with less weight - my 140 pounds wasn't really enough. But that may be out of date, so I won't list the brand.
Regardless, if you buy any device, make sure it is rated to take your personal weight. Some of the cheap exercise and dance spin or twist training devices weren't strong enough for my 140 pounds.
After over 20 years, I still get dizzy trying to spin on the ice. I think the best advice for dizziness I got was to try to ignore the dizziness. I get more dizzy if I look up, like people do for a layback spin. Though I wonder if that is partly because I lack the flexibility to look all the way up. But other people with more flexibility I've talked to said looking up makes them get dizzier too.
Based on the idea that dizziness is the result of confusion between what the eyes see and what the balance organs in your inner ear feel, I tried closing my eyes during skins, but at least for me, that is counterproductive. For me, eyesight is a huge part of balance. But it is an interesting exercise, that might help, off ice, or on very uncrowded ice. (Sometimes you can be alone on the ice, at some rinks, if you come to a mid-day public session during the first few minutes of the session.)
Ballet people have an interesting approach to reducing dizziness called "spotting". They continuously turn their heads to keep their eyes on a distant point. When they reach the limit of neck twist flexibility, they snap it back around in the other direction, much like an owl can do. But ballet turns are a lot slower than good ice skating spins. Even ballet people dancers sometimes get whiplash doing it, if they don't train their neck and nearby muscles enough, and you easily could get whiplash at the faster speeds of a good skating spin. (In fact, the physics is such that spotting slows down your average rotation rate.) But a few figure skaters who had serious ballet backgrounds have been famous for doing spotted spins.
But it might (or might not) help you to concentrate on one distant object, without turning your neck, as you come around opposite it. Some people say you should look
slightly up - e.g., if there is a clock on the wall.
As I've gotten older, my spin balance has gotten worse. I only try to spin slowly now, so I don't try to bring my arms all the way in. My best spins are often when I try to twizzle, and the closest I come to twizzles are often when I try to spin. Which says I'm dong something wrong! I asked for help from many coaches and other skaters. Some of them helped me a little. But I think spin balance is individual enough, that you may want to try several coaches and several approaches until you find one that works best for you.
Rocking forward and back because I can't feel I am on a constant point of contact is one of my biggest problems. I don't know if it is an issue for you.
Playing with different skate blade shapes helps. Specifically, some say that shorter radius main rocker length, and even shorter radius spin rocker length, is best. The closest I've come is with blades that have a well defined (abrupt) transition between main and spin rocker. (Less than stellar skate techs tend to round out the transition, so you can't feel when you are on the transition point, that should, according to most coaches, be close to underneath the ball of the foot (which you can do by slightly customizing the shape of the blade(s), or by moving the blade mounting point. The failure of techs to keep the transition abrupt is one of the reasons I switched to sharpening my own blades.)
I've tried creating a very slight direction change at the transition point, sort of like a top, but am not sure that actually helped. I think it did, but am not certain. The biggest thing is to make the transition abrupt, so you know when you are on that point. Or to do true scratch spins, where you can feel the toepick lightly brushing against the ice.
As blades get older, if the skate tech doesn't trim back the drag pick to match, the length of the blade that can touch without the back-most toe pick touching gets shorter, because of the metal removed from the blade during sharpening. That might make it harder for you, and force you to spin farther back on the blade as well. If you have a really good skate tech, consider having them trim the toe pick. But if you don't, they could totally destroy the toepick. Using a really first class skate tech, who can watch you try to spin, so they can adjust the blade to your use, or learning to sharpen yourself, might really be one of the best things for spinning and twizzeling you can do. (But as with spinning, sharpening has a learning curve.)
One thing that has helped me a lot was in fact moving the mounting point so the transition point was underneath the ball of my foot, like I mentioned above. But also, to move the blade to the left-right center of the skate. I once had sideways-offset blades so I wasn't on one edge all the time. I changed to move them back to center, and instead changing the shape of my insoles to give me the same balance fix. Again a first class skate tech can help you do this. If the new mounting holes overlap with the old ones, or if you don't want water to get into your boots, you (or, better, the tech, if they are any good) need to first fill the old holes with something - e.g., a thinned dowel rod, or leather strips, that you hammer in. I've also used shoe goo, but it takes a long time to set, though it is faster if you heat it. (I've only tried such things in leather soles. Boots with synthetic soles likely need different fill methods. E.g., Edea provides special hardware to fill holes in Edea soles.) I wonder if fast setting epoxy works. I'm not sure if you can drill new strong screw holes through it.
Honestly, in addition to that, the closest I have come on or off ice is by spinning in alternate directions. One spin clockwise, one spin on the other foot counterclockwise. Repeat forever. Somehow alternate spins help me regain my sense of balance, and reduces dizziness a lot. That helps for twizzles too. You may spin better in one direction, and alternating is usually taught only as an advanced skill, but try alternating to help for the sake of dizziness and balance anyway.
Of course, you need to transfer the direction of your spin axis to match the foot you are on. If you like physics, AFAICT, the idea on ice is to make it so the axis matches your effective vertical, meaning the vector sum of gravity and centrifugal force from skating around the small circle, and should pass through your center of mass. (I think. Maybe it's really a weighted average of center of mass and center of gravity, though people are short enough they are almost the same.) Any other axis would precess. Though of course moving arms might throw off the ideal axis a little too.
Try different arm positions. To the sides at first, above your head (I think keeping a vertical stretch through my whole body might help me, but maybe that just helps me align body parts), at tummy level, behind your body. Depending on your body symmetry and other things, you might not find matching arm positions balances you best. I come closest with one arm to the sides, gradually moving inwards, while the arm on the side I'm spinning round stays raised vertically in the air. I also like one arm across my tummy, and the other arm behind the small of my back. Of course different arm positions create different balance points, and require different spin axis.
Overall, you may need to experiment a lot. And find the coach who helps you personally spin best. It might not be the coach who helps you best do other skills - though it is impolite not to tell your main coach you are trying someone else, even if just for one skill. Some coaches even specialize in teaching spins. Perhaps your main coach could suggest another coach who does just that. Be careful - don't be surprised if your main coach is offended by the idea you would try another coach, even for just one skill. I suppose you could use videos, but getting feedback on your alignment, etc., from an experienced coach is a big deal.
But playing with spinners and twist trainers are worth experimenting with too. But I suggest you start with the cheap and more stable exercise ones. They are cheap enough not to matter much if they don't work. Or with slippery socks on polished or waxed wood or waxed plastic floors. Which are even cheaper. And remember what I said about practicing falls and wearing skin-covering clothing for safety.
