Strong flat magnets at the tops of the doors where they hit the jamb.
Or, once you have resolved the door opening problem, you could mount some nice brass kickplates vertically on the doors next to the opening, which is the place the cats are presumably accustomed to manipulating to get inside. They won’t be able to damage them, and the frustration of wearing out their claws there should soon discourage the activity.
Makes sense. Actually our bedroom doors had that problem too when we first bought the house. You had to push the door up/straight so that the top bolt would latch, otherwise it sagged too much. The problem was simple: someone had screwed the door hinges into the frame with inappropriately small screws for what are relatively heavy solid doors. Appropriate hardware fixed the problem.
I think this might be the best option. I believe it would work to stop the cat, and would not deface the door or risk someone getting locked inside.