sorry for what your going through
Your children have a stable home with YOU, their home is you and wherever you are, he is nothing but a fixture that can change as you see fit. Yes that's a terribly cold and emtionless thing to say about him, but this is the way he has treated you...carelessly and without much thought because he prioritised a bit on an office flirtation, sexting and getting his end away over YOU and his family life with you.
Yes maybe he has been stupid, maybe it's a one off...but do you really want to hang around to find out? Wait and see if he does it again?
Your in a good position to leave if you want to, you absolutely do not have to put up with him treating you this way. Unless you truly believe it was a mistake and that he is genuinely sorry and will never ever doing again, I would rid myself of him. Living in a relationship where your looking for clues, wondering if he's telling the truth or lying and hiding things does nothing but absolutely and utterly destroy your self confidence your self worth and leaves you a mess. Better to leave while your strong and yes while your self esteem has taken quiet a knock your integrity and dignity will be well and truly intact.
OP: whether or not he is genuinely sorry, whether he is worth the trouble and the hurt of trying to recover from this betrayal only you can know as only you know him. I do know that worth it or not, staying is often much harder than and I leaving. You have to figure out if he is worth staying for, for you.