Hi!
I can’t personally speak to how a bunny reacts after moving, but I have a few ideas on potential factors in the change of behaviour. Of course, your bunny could still be exploring and is very slowly warming up to the new house. My best guess is that the floors are a little bit more slippery in your new house, and that’s contributing to your bunny’s tentativeness. I could totally be wrong, but whenever I’ve let Paisley out on our wood floors, she acts very nervous and careful, sticking close to the ground and moving very slowly. It could just be that your bunny isn’t super comfortable getting around on a more slippery floor. There could also be other factors like noises, smells, etc that may contribute to the avoidance of the living room and such. I’m just guessing here though
If it’s the flooring, my only suggestion would be to maybe add some carpets? Or fleece blankets etc. I know this isn’t always the most visually appealing, but there are definitely ways to incorporate some flooring with easier grip.
Hope that helps! Sorry if I’m totally off haha