That's my first post, I have been reading your posts and others since a few months now. We have a lot in common that's why I became attached to your thread.
You are a very courageous woman but your husband is not on the same planet as you are in term of logic, he is still in the fog, he is still missing his AP. He cheated because something is wrong with him not you, even if he trying to convince you of the opposite, he needs to find a excuse to justify his behavior and his actions because he knows deep down that's plain wrong to cheat and he is provoking you with his remarks to push you to leave him so the decision won't be his but yours, another trick to shift the blame of that kind of decision from him to you. He still believes the world revolves around him and his only priority is his own happiness. It is a strong tendency among cheating male physicians since they are used to be served and their egos stroked several times a day by a myriad of women, I am married to one (powerful and successful) and I witnessed also first hand how they can be nasty with with their wives when they want out.
I know how much pain you have in your heart and how much it hurts thinking your children might grow without having their father in their daily life. They need you as a strong loving mom and they need you not to be in a constant internal pain, and right now you might need to let it go (GAL) not only for you but for them. Even if they are still young they do feel that something is not right now (my youngest one was very sensitive to the tension in the house, the oldest ones they knew what he did) and ask him (verbally) perhaps his opinion about him moving out for a few weeks or months (also in case of divorce he was the one who abandoned the marital home, better to secure full custody of the kids, you might be really surprised how some who didn't want the kids fought for them just to keep some money or put pressure to get what they wanted on their wives) so he can make up his mind without pressure and then you will evaluate the situation (No contact others than the ones needed for the kids, also keep all those messages for later on, it can show how much involved he was with the kids), you need to find some rest, some peace, your home should be your refuge so you can recharge your battery after you workday and be yourself for your kids.
It doesn't mean you have to ask for a divorce but it is obvious you tried everything and you need now to focus on yourself and your kids. It took me several months and the help of a few friends who went though that ordeal to realize that he was only focused on himself after that epiphany my life changed, I started to detach myself from him and my pain went down significantly, it doesn't mean I didn't have any more feelings for him but my mind was not obsessing anymore on how to fix him or our relationship, I was able to rebuild myself as a person and a better mom, they only had me since he didn't care much about them, and college was around the corner for 2 of them. Since we (me and the kids) detached, he is becoming more interested in their lives.
Also boundaries need to be established and reinforced, being respected is a must, be specific and state the consequences. Remember children learn life and what is ok or not by watching us. At one point my son started to talk to me the same way his dad was, that's when I realized how kids can reproduce relationship models without being conscious of them, since the boundaries are up he changed because the boundaries are for all of us.
You will never be able to convince him to change in his current state of mind, the more you will chase him the more he will run away. He never had to be afraid to lose you since you kept begging him, he was always in charge of the game. Remember cheaters like to be in control, because that their way to feel in power/secure and they will say whatever you want to hear (don't believe their words) to stay in control of the situation. When they feel that they are losing you they become nice just to give you some hope to regain control on you. Also once he will be living by himself he won't be able to blame you for whatever is wrong in his life, also he will realize that taking care of kids is really time consuming. Sara, you need to asses what is the best for you in that very moment, it doesn't mean the decision you make will have to stay for forever, you can always assess, change, re-evaluate.
Life is not a straight line, it is more like Honors Geometry, several steps before the final solution and sometimes you might have to recalculate again specially if the professor decides to change the values.
Plenty of hugs,
Me 52+ WH 57+ Married 20 + Piecing since 2016 (ready to give up...)