Dear Julia,
I really understand your problem!! Know how terrible it is to decide between your beloved pet and your health because I found myself in a similar situation with my pet hamsters. Here is some advice that I hope will be of help to you.
First of alt, do you have any friends or neighbors who would like a pet? If you do, you could try giving your cat to them. If you can do that, you will know where she is and that she is being treated well.
I also think it would be a good idea to get another pet, an animal that you aren't allergic to, like a fish or a turtle maybe. That way you might not feel so sad and lonely when your cat is gone.
I hope I've been able to give you a few ideas.
Good luck!
Andrew