I impulsively bought a cat when I've never had a pet before

I can't keep him. It's not due to lack of funds, the fact that he's sick while rejecting medicine, that I always have to clean after him. I think I'm allergic to cats. Ever since I got him, my throat, nose, eyes, ears, and skin has become itchy. My nose is congested, my eyes are watery, I’m wheezing more often, and because of this I’m having more difficulty sleeping. I feel better when I take decongestant medicine but I also feel better when I’m not around him. I hope I’m right so I can get rid of him without guilt. Though my family’s already really attached to him and I need their funds for an allergy test, so I don’t know what will happen. It’s sort of funny, because I’ve grown up in a toxic household and I planned to leave after my college graduation. But now that I have a cat my family oh-so-adores, I feel like I just trapped myself. Can someone give me some advice or something?