OP, I'm pretty much in the exact same situation, 3 young children, been together 20years and found out my DH was having an EA with his cousins girlfriend who lives overseas. We had drifted apart and become parents and stopped being a couple, I thought this was the way it was and just thought Oh well, this was the life I chose and I didn't pay him much notice to how unhappy he was or him to how unhappy I was.
His was all online and like yours vomit inducing gushy messages, and after I found out tried to keep in contact for another week or 2. The lies and tricky truth went on for a good month after he told her it's over, this was Nov last year so it's been about 9months since I found out.
In the beginning he would get very defensive if I asked questions but he started counseling and we did couples counseling, he gave me total transparency, access to everything,
Blocked her on everything. Maybe about 2 months after finding out things started to improve and we started to re-connect.
He was completely different, would bring up the affair when he could see I was anxious, started making efforts to go out together just the 2 of us and try to make me feel like I am important to him.
I won't lie I still totally snoop through his emails, what's app, Facebook messenger, and everything else, but not nearly as often as I did in the beginning maybe once every week or two, seeing absolutely nothing and no signs of any contact everytime has helped to re-assure me it's over.
For me it's my head when I wake up or trying to get to sleep that pictures the messages I saw, or the little lies he told or the half truths where I wonder do I know the whole story and that play though my head at 3am not letting me sleep. This is not every night but maybe when he a not giving me as much attention (I'm a bit extra needy because of this) he knows and if I tell him he feel distant he makes an extra effort, it happens maybe once a fortnight or so but for a coupe of days till I tell him everything I'm feeling and get it out in the open.
I did find early in journaling really helped, I had such bad anxiety from it, also a calming app of ocean sounds when I can't sleep and when I had intrusive thoughts, I did as my therapist told me, acknowledge them, it's your mind trying to protect you from danger. And I say in my head, ok mind, thank you, I'm all good right now and push the thought away, sometimes it took a few repeats !
My posts/story are on here if you want to see what I was like at the time !
Just take it day by day, it does get easier but I think like someone said it can take 2-5years to really heal and trust your gut !