Sorry to hear about this mess that you find yourself in now.
First, he needs to make a choice. You need to make him make that choice. Don't let him have it both ways. I think if you continue to let him vacillate in this decision, the harder it is going to be on you and the kids. Honestly, it's a bunch of crp that he packed his stuff up and moved in with her, next door, but still says he loves you.
If you love him, if you feel you can get past this, and you feel you can fight for him, then do it, but I'm thinking cut your losses. Let him believe the whole grass is greener thing and you do what you can to be strong for yourself and children.
Regardless if he comes back to you or not, you need to find another place to live as a family or just you and the kids. Having you both next door to each other is sickeningly convenient for him. Regardless if he comes back to you or not, you need to find another place to live as a family or just you and the kids.