Giving away my cat but I don’t know where to start

Honestly this pains me so much to do🥲🥲 I don’t want to rant here but in short: I’ve had my cat for almost a decade and I need to give her away due to an unexpected adoption of a dog (almost two years ago) that might as well be impossible to get the two along, causing me to have to keep my cat cooped up in my room. It’s driving me crazy and it’s ruining my own mental health which lead up to not having the best care of my cat (I feel extremely guilty) But now I’ve finally convinced my mom to start looking into rehoming her

Anyway, obviously I don’t want to give her away to just anyone.. but also giving her away won’t be easy. She’s a senior cat, overweight, and is extremely paranoid of other people and animals. I want to give her away to someone who is willing to bring her back to health and give her a home she deserves to be in but I don’t know where to start and im so paranoid of the possibility of losing her to a bad person. I live near Houston Texas if that helps but yeah

Any advice would be appreciated and if u want more info on her life I’ll reply