Cat Rehoming In Houston?

I have a cat that I'm considering rehoming as I can no longer take care of her. Are there any places in Houston that let you surrender the cat but you can still see the cat from time to time? I would like to still see her sometimes, I just can't care for her. I'd prefer giving her to a regular person. I feel like shelters euthanize cats because there are so many of them. Any suggestions?