How to find another home for my aging cat?

We have a pet cat at home, I believe he’s an American shorthair, ~11 years old. We got him ~5 years ago because my sister wanted a pet cat. The problem is that my sister somehow developed an allergy to him and she has a lot of trouble breathing in his presence. The cat is also allergic to something unknown so he keeps scratching himself, causing a permanent pinkish scar near his ear. The only way to stop it is if he has a cone on. Whenever we take the cone off, he immediately begins scratching again. We brought him to the vet and she prescribed some sort of topical solution to put on his scratches but nothing has helped. He has had this problem for almost two years now.

This cat also screams every morning at 6am for food for some reason, waking my mom up. My mom already doesn’t sleep well, and this doesn’t help. We feed him a lot so idk why he believes that he’ll start starving if he doesn’t act like a morning alarm.

We came back from a month long trip a few days ago and for some reason, the day after we came back, he peed on my sister’s stuff. It was A LOT of pee. He had no problem using the litter box in the past, so idk why this happened. He’s also screaming a lot more now. While we were gone traveling, we had a friend feed him and clean up after him, so he wasn’t neglected or anything.

My sister will be moving soon and she can’t take him. I’ll also be moving soon, and I will not have the time to take care of him nor do I want to. My mom is getting older and doesn’t have the energy to continue taking care of an aging, sickly, cat.

We’ve already contacted some animal shelters earlier this year, but they’re all full. Does anyone have any ideas on finding people to take this cat? I don’t like the idea of leaving him to a stranger, but we are at our wits end.