Well, for what it's worth it sounds like what you need to do is not throw away what could be a really good relationship, but to learn to trust. There is no reason why this shouldn't be the right man for you and your kids. He has cheated once and owned up and may never do so again. Or, he might. But throwing away a good relationship on the basis of a 'what if' seems like a sad way to live. My DH might leave me one day, I hope to hell he won't, but I'd so much rather have my marriage than be alone just in case my other half leaves/ gets ill/ gets runover etc. (There are lots of different ways of leaving!)
I don't have kids so I can't begin to imagine what's best for them. They are of course the most important people in all of this but I'm sure others with more experience can make suggestions there.
But looking again at your letter, you admit that you think you are a bit paranoid, and worrying about him texting other people - why not tackle this before you throw this relationship away? If you can learn to trust, it'll help you in this and in any other future relationship - including that of you and your kids. If the trust is repayed, there is happiness. And in my experience most trust is rewarded and actually paranoia and mistrust can do more to damage a relationship than a lot of things. And it'll make you really unhappy.
Let's face it - we all have friends who are men and women and personally I'd feel gutted if my husband didn't trust me to text my friends, male or female. I stay away for work and often meet and go out with my male friends. I love their company, but it would never occur to me to risk my homelife for a tumble with them (ew!)
Perhaps you could sit him down and say you know that from the past that you have trust issues and you know this. But you take what he says at face value and you really cherish your relationship. If he cheats, you will leave him. But you don't want to end anything so you need to work together as a team on this, to keep you and your little ones happy. Can he help you to do this? If he's the man you hope he is, then he will, and you will have hopefully properly then got over the enormous and horrible betrayal of your husband. Don't let that ruin the rest of your life!