littlebear, firstly, huge hugs. Sounds like you are all going through a huge transition.
Listen to your instincts and trust them. Perhaps in the short term they are better spending most nights at the place they call home? Your split sounds amicable, are you able to spend time with the kids in their familiar surroundings until you get settled into a new place? I think the kids should stay together as much as possible. Who is normally the primary carer they are used to? If it's you but in "his" house, would your ex consider letting you stay there for the sake of the kids? I'll try to think of some other stuff...