Oh, M. J., I am so sorry - how rotten and unfair.
But, don't walk away from anything - his affair is ending the marriage and he should be the one to leave. Also, you don't want him to have any ammunition against you when you go to court.
So, make him move out, change your locks, notify the schools that he no longer lives in the home, call an attorney, open your own bank account and transfer as much money from the household accounts that you possibly can.
As for the mortgage, that will probably be settled with the divorce decree. He could quit-claim his portion of the home to you, you could refinance the home into you name alone, the home could be sold with the proceeds divided between the two of you. There are many options. The easiest is to have him quit claim it to you, leave the mortgage in place, and you continue paying it. This is what my sister has done with her home when she was divorced because of his affair.
Most of all, be strong and calm for your children. They will have lots of questions and confusion when Dad moves out so think about what you will say to them beforehand so you are prepared with adequate answers.
Childcare - depending on the ages of your children - check into after school care at their current schools - that way no transition to another facility after school. If they are younger than school age what about an in-home sitter for them (one who comes to your house)? Always, do spot checks on the facilities you are considering - when I had to pick one for my son I just dropped in unannounced a few times at different times of day to see what the atmosphere and activities where like.
Stay strong,
Good Luck and God Bless