Hi and welcome to PT.

Rather than give the cat away, you may want to ask your vet for some mediation to keep the cat calm. It often gives them the time to relax and become accustomed to their new home. Some people do not feel comfortable doing the medication thing, but it has worked for my household in the past. And then, it didn't work for another time.......

Good luck.

By the way, new homes are sometimes the only answer. I have re-homed a cat and boy is everyone happy now. You need to find JUST the right home though........

SAS