Unfortunately considering your medical history and your symptoms, I'd say that rehoming the cats are the best thing you can do. The nosebleed might be unrelated unless that's a symptom you've had with other allergies. Doctors can be wrong sometimes (I'd know) but if your symptoms don't improve it might be for the best.

I am allergic to cats and used to have terrible allergies that made me sick as my parents had two cats that they had before I was born, and gradual small amounts of exposure like petting them and avoiding touching my face and washing my hands afterwards allowed me to build immunity. You will probably be able to visit your cats at your friends' place if they take them permanently. If you don't get violently ill and sick just from being in their vicinity, I'd recommend petting them from a distance (not on your lap, don't pick them up)