Emily, I can completely understand where you're coming from. I started skating when I was older, after years of dance and gymnastics, and after a few months I was about at the point you're at and also struggling with spins. The speed of the spin and the fact that it's too fast to spot had me on my back side a lot for the first month, and even after that I thought I'd never be a great spinner, but with time it's become one of my favorite parts of skating. Something to remember here is that skating, though it has many similarities to dance, still has very different biomechanics and physical forces, and it will take time for your mind to figure out that you have to move your muscles differently than you do when you dance.
With eleven years of ballet, I imagine that you are used to brining your toe to your knee and maintaining turn out during pirouettes. This won't work the same in skating, and could possibly pull you off your center. Your coach has probably covered the proper leg position, and please take his/her advice before mine, but it helps to remember that scratch spins pass through an "h" position instead, with the knee facing forward, and the foot pointed straight down to the ice, not brought in towards the body. After this "h" position is attained and you still have your center, then you can bring your foot around your leg and pull in more easily. Additionally, in the long run, this will give you more centrifical force when you do pull in. Perhaps you can practice centering your spins before pulling in, and then only trying to cross the legs when you're comfortable on your center. After you've started your spin from whatever entrance you're comfortable with, try centering with your arms in second position, and your free leg mirroring the arm above it, extended to start. Once you're comfortable there, bring your arms to fifth en avant and your free leg to the "h" position. Make sure your knee is bent to only 90 degrees at this point, no more. Work on centering the spin in this "h" position. Once you're comfortable there, think of bringing your knee slightly across your body before pulling in. This turning in of the leg may feel awkward at first. Pull in only so far as you feel you can with control. Eventually you'll be able to bring your leg all the way across and slide your free foot down towards your ankle, but don't rush it. Though there are similarities between your center in dance and your center in skating, there are also some differences that you need to get used to.
Another thing that might help is allowing your skating knee to stay slightly soft instead of pulling up through the knee as you would in ballet. This slight softness is very subtle, not even really a bend, but it's important to centering your spins. Also, if you're entering your spin from the tradition spin entrance with a step forward to outside edge, be sure you're stepping "into" the circle you were on on your backward inside edge before the step, and not behind . You'll make more of a 90 degree angle and should feel you're momentum is going forward as you push into the spin, and not sideways. This will allow you to curl that edge into the spin a bit tighter, and thus your spin will be tighter and more centered after the 3-turn that starts it.
I've used all of these techniques and more to overcome dancing habits and make my skating spins better. If I'm working on my spin entrances, I'll think something along the lines of "into the circle, tight edge, soft knee, second position, h, pull-in" as a way of giving myself mental cues that I can attatch to the ideas of what my muscles should be doing.
Some of this can be practiced off-ice as well. The movement from the open/second position into the "h" position can be done in tennis shoes, and this exercise might help if you find yourself subconciously trying to maintain turn-out into your spins on ice. At the same time, you can practice keeping your standing/skating knee softer.
I often get frustrated on the ice, too, although I've gotten better about that with time. Coming from a background where dance movements seemed to come more naturally than skating movements, I expected skating to be less foreign to me. My first coach was constantly reminding me that skating is not a natural motion, and that my body will need time to learn the totally new muscle memory it needs. Nothing else we do- dancing, walking, even just holding a position on the floor- requires us to cope with the fact that we are on a moving blade. Our bodies intrinsically react to that strange new motion in ways they would have if they were stationary, and this just doesn't work; as such, we have to learn to move in a new way. My coach reminded me of this again and again as he could see me getting frustrated that I couldn't just do everything right away. Also, I had hoped that my initial pace of progress would be maintained through higher levels as well, but I've found that that thinking was completely unrealistic. I'd love to believe that I could have gotten doubles over a summer of work like Casey did in the movie "Ice Princess," but in reality it takes most people, young and old, years to master singles and basic doubles and the spins and steps that go with those levels. I've had to learn I can't force it, as much as I really want to, and that I need to instead focus on what I need to work on instead of what I think I should be doing. It's not easy for me, and I still get frustrated to the point of stomping the ice and grumbling at myself, but I find that if I focus on technical things to work on I'm a bit better with solo practice.
I don't know if that all helps, but I hope it does.
Thanks for joining us. Post often, post long! 
