Hope you're ok OP. I have been through this. Except I was very naive and believed it was just messages, stayed longer, and wish I hadn't.
When men cheat, in my experience, they will only ever admit to what they've been caught out doing.
If you see text messages - I only sent a few harmless messages, there's nothing going on, honestly.
If you have a friend who see's them in town - I literally just bumped into her, she was a bit down so we went for a quick drink, nothing happened I promise.
If you catch them on an online site - I was only looking, there's nothing going on, I would never meet up with anyone on there, honest.
If you actually catch them in bed together/or see them together and it's obvious what's going on - it was just the one time, it has never happened before, it will never happen again.
You think by them admitting it, and 'telling the truth' that they are somehow worth forgiving. But they're not. Because they will only ever admit (by force) from whatever evidence you have gathered. And if all you have gathered is a few messages, then they do not have to admit to the rest of the story. And they won't ever give you more information than you need, so they can convince you you're the one being unreasonable, that if you can't forgive a few messages, then you're the one with the problems.
Then of course they will roll out the usual if we were 100% happy, then I wouldn't be sending messages would I? Please don't allow him to turn this back on you.
Also, he hasn't just betrayed you, but your children as well.
Please be strong, please don't fall for his excuses. No marriage is 100% perfect, that is a given, but crossing the line like this, I personally would never forgive.
Good luck OP xxx