Sorry to hear things have gone downhill. Is he willing to write this woman a no contact letter with you seeing it and the response? Is he willing to be completely transparent and give you passwords to all his email accounts? What does he want? Because in most states men are limited to just one wife. Retrouvaille could help you if he gets rid of the OW first.
If he is not getting rid of the OW, then I think you do need to tell him to find somewhere else to live. Being married and raising children does not go well with dating.
