If it is a specific breed cat, you can get the breed rescue cat organizations involved. Any possibility you could take the cat and foster it while you look for a home for it yourself? Sounds like it might be running out of time.
People are so hideous when they decide to put a cat to sleep for selfish reasons, such as they just don't have time for it, can't keep it and are to lazy to look for a home for it, or when it becomes a behavior issue and they just don't want to deal with solving it- so hey- just put the cat to sleep- I'll get another one down the road when it is more convenient!
I wish those types of people OFF the planet!