I am allergic to cats, and I live with three of them. I love cats and wasn’t always allergic, but for some reason I now get the sniffles, sneezes and itchy eyes when it excessive contact with cat dander. When I go home for the summer to my house full of cat dander and dust, I’m usually very sneezy for the first week or so (or whenever I clean) but my body seems to get accustomed to it after a while and I can function quite normally.
If you do choose to live in that apartment with the cats, there are a number of things you can do to avoid having too much trouble with your allergy symptoms. Never let them in your bedroom, and change your sheets (especially pillow case) frequently. Take non-drowsy allergy pills like Claratin every day. Go to an doctor who specializes in allergies; some doctors do allergy treatments by testing what you are allergic to and giving you weekly shots to gradually increase your tolerance for those allergens. It’s a process that takes a few months, but it can be well worth it, especially if you do go for that living situation.
However, it does seem like your cat allergies are fairly severe compared to mine, especially if you have trouble breathing. Would it be possible for you to spend a day or two in the house before making a decision, to see how badly your allergies react?