You say your cat isn’t unwell, but several things suggest otherwise, including excessive urination, a clicking hip and poor mobility, general unhappiness. Yes, she is elderly, but it does sound as though she’s not enjoying her old age.
On the other hand there’s your anxiety and concern for her, and the distress the whole situation is causing you.
Then there is the question of rehoming an elderly cat with hygiene, possibly health, problems. That doesn’t sound like the kindest solution for a very elderly cat which has been rehomed at least once before.
If it was my cat, or dog, I’d take him/her to the vet for a health check and be prepared to face having to say goodbye. It might be the kindest option, all things considered.