You might want to get some legal advice on your question, but from my experience in Calif. you are not giving up rights to house (community property) or custody since it would be a mutual agreement. None of this will be easy for the kids, but having them live in one house is better than going back and forth.
Do you know what your H hopes to accomplish by moving out? What do you hope to accomplish during this time?
The task ahead of you is never as great as the Power behind you ~ unknown