One of my cats was an avid screen climber. We'd find her at the top of the sliding screen door, just dangling there, quite calm and pleased with herself.
We found stronger screens (I can't recall if they were specific 'cat screens') but they were advertised as being pet-proof). I think you can find them at Home Depot or Lowe's.
I didnt' know about Feliway at the time either but that might help--a spray or a diffuser near the worst spots might calm kitty enough to not care about screen climbing. What about the bitter sprays? I realize some will go right out the screen but enough should linger on the screen too. New smells outside are probably overwhelming. YOu could also try securing foil or bubble wrap where kitty tries to climb and even though that might take some repeated sessions, that could modify the behavior. I also placed one of my tall cat towers near the screen door so she could do her manic climbing and still see outside.
I am sure it's all just new to your cat and with some patience/time, things will wind down.
Moves are challenging; we have moved multiple times with multiple cats. My cats have been amazingly flexible about it all. Did you isolate your cats to one area for a bit then slowly let them explore larger areas? We are moving again in a couple months and I am already strategizing