Since you are noticing she is less social with you, and more so with your husband and son, I wonder if there could be a couple of things at play to consider.
1. You are the one that is transporting her/picking her up etc. She may see that as a negative since most bunnies don’t enjoy being picked up, even if they tolerate it well. I know it’s something you have to do, but maybe you could offer her a treat right after or during the picking her up.
2. Scents/smells can play a part in getting the ignored by a bunny. So think of some possible differences that you bring — cosmetics, shampoos, body lotions or perfumes? Also, not sure what the family duties are but if you do more cleaning and cooking then the scent of cleaner or food might be on your hands. I am sure you wash your hands, but even then, what is the scent of the hand wash? (NOT trying to be stereotypical here – sounds like I am thinking we women only wear make up, wash our hair, wash dishes and cook LOL!) Anyway. Just some thoughts to ponder.
Regarding litter training when moving to different location — try and keep a set-up someone the same — even if you have to create a base set-up in both places. So her bed and litterbox are in the same place within a pen. And then when you move her from one location to the next, don’t give her free time yet until she has had time to settle into her home base.
Regarding giving her things to do — she is still rather young at only 13 months and it sounds like she does enjoy exploring. After she has settled into her home base, and you feel okay with letting her run about, get her boxes and/or tunnels to run about in. And then switch it up every couple of days. Hide a treat somewhere in the boxes and tunnels. This is enriching for her and will encourage her to exercise. My bunnies who reached over 5 years old definitely slowed down, but a new box or tunnel always got their curiosity going. Even now, Vivian who is around 10 now, blind in one eye and much less active always has to check out the new box or tunnel nearby.