Rehoming a cat

I never thought I’d be writing this but I think I need to rehome a cat. I have a 12 year old cat who I’ve had since she was a kitten and is pretty much the love of my life, but she’s old and while she is very tolerant of my kids, she isn’t too thrilled with the attention they give her. So a year ago I decided it was time to get a second cat, a younger one that was smaller so the kids could pick her up and had lots of energy, and who could take the attention off my older cat. I found a kitten that had lived with a 6 year old who dressed her up in doll clothes and was good with kids. When I brought her home my old kitty was not happy but she had lived temporarily with other cats before and always ended up getting along with them so I thought if I did the slow transition and followed all the advice things would resolve.

A year in, she and the kitten, who is a young adult now, still don’t get along. I feel like I’ve tried everything. Multiple sprays for the rooms, keeping them as separate as possible, slow transition, I even hired a cat behaviorist and followed all of her advice and nothing has worked. The new cat still attacks my old cat, which is causing my old cat a lot of stress. As a result, I have not bonded with the new cat, and my 3 and 5 year old really don’t give her as much attention as I expected them to so I feel like she’s just not a good fit and I don’t think it’s fair to my old cat to keep her around.

I’ve never had to rehome an animal so I don’t know how to go about doing this responsibly. The new cat really isn’t a bad cat. She loves attention and wishes I would give her more, she is really good with the kids (lets them pick her up and play with her and never bites them), and I think she would get along well with another cat that wasn’t as old as my cat and maybe a little less docile. How does one go about rehoming a cat when they don’t have friends or relatives that can take her?