Hi Angela,
I think your fear (or possible) fear as you describe it more widely seen then one might think. Yet to conquer a fear you have to overcome it by doing it. I was never a confined spaces person, but I find more that I have dove the better it has got. (Swam through a few wrecks, etc.) I think the best thing is to have a good instructor, and then go with people who are experienced and those you trust. I have dove with a few people (not ice diving) once and said forget that. Then on the other hand dove with others that I wish I could dive every time with. Remember a good instructor won’t let students panic or “flip-out” as so nicely put by others. A good instructor will see it coming and comfort (for lack of better words) the student to ensure panic or fear doesn’t set in underwater.
That being said, I say go for it. I intend to sign up for ice diving this winter, as I think it would be awesome! The few people I dive with I haven’t yet to twist their arm to do so, but time will tell. Where are you thinking of taking the classes?