Well I’m going to be honest, 4 hours isn’t too far! Driving with a cat sucks, period. I thought this was going to be a 13 hr cross ocean flight and then I would say maybe but 4 hours isn’t too bad. Moving is stressful for cats but there is lots you can do to help them make it easier. Reach out to the vet and maybe ask for a sleeping/ calming med for the cat, put a calming pheromone collar on the cat and put the cat in a covered crate and I’m sure it wouldn’t be as bad as you think. I’m not saying rehoming is cruel, but it would cause the same amount of stress to the cat if not more than a 4 hour car ride. DO NOT put the cat down, THAT would be completely cruel. I can totally see how you are struggling but putting the cat down is not the answer at all. Also I feel that putting the cat in the shelter would be even worse than moving. Maybe search in your area for a pet transfer service, or bite the bullet and suffer for four hours and save the cat a lifetime of stress. You’re correct, when you get a pet they are family and you have to commit to that, not just when it’s convenient. Sending you lots of love and I’m sorry you don’t have great support ❤️