Much love to you, if I could live everyday trying to save every person from dealing with this and feeling this pain I absolutely would.
I'll reiterate what Chamomile says here, you need to call the shots, it's not time for the pick-me dance. YOU are the morally correct person, you stand on the side with the rest of us who have been cheated on that chose to do right for ourselves, our children and to stand respected.
Repeat the above to yourself, you are devastated but you are morally correct. You will never have the dark cloud over your reputation and your kids will always see that and respect that.
Just as Numis67 says, you will come out of this stronger, after my first WH's round of cheating in 2015 I was a wreck, it took awhile but my now recent discovery after reconciling 6 years later, yes I fell apart for a few days but I am moving forward with divorce. I'm putting the pain into building a life without someone who has ZERO remorse or care for anyone but themselves.
Your girls (I have 4 daughters) need to see you not accept this behavior as okay. Even if they're small and you go your own way in this relationship & understand later what mom had to do, they will see this as unacceptable in their own future.
Yes, they always say, we were just friends, lie and say it wasn't intimate, it meant nothing, you were too busy for me (with his kids of course), you did that one thing to me in the past, they'll try and blame you over and over until they think you believe it and THEN beg for you give them one more chance. This^^^ is the cheaters playbook to shifting the blame.
Cheating isn't about upgrading with AP's (affair partners) it's about filing the empty void in a morally corrupted person's self. My WH didn't discriminate, if he could get them into sex that's all it was about, for my WH it was about the filling of his low-self esteem by thinking he was "attractive" enough to get something sexual off of someone.
Stay safe and strong, you are the prize in whatever you decide.