Well, I don’t think he handled this well, but it all sounds pretty legit. He should have told you he was communicating with her for closure. Hiding it from you seems unfair, but he probably figured you’d get upset. Now that you’ve found out, keeping it a secret has just made things worse. It looks shady now, when he could have been straight up with you about his guilt over the break-up.
Did he hide the fact that he’s married so that his ex would talk to him? If so, that seems a bit much. Does he tell her he loves her in the texts, or do you mean he told you that he once loved her? There’s a huge difference between saying he was once in love with her and writing “I still love you” in a text.
We all have baggage. He was carrying around guilt over this relationship and needed to let it go. Would it have been more mature to tell you about it and ask your opinion before texting her behind your back? Yes. Should he have deleted her phone number? Probably. But if letting go of his past is going to help your marriage, a few texts seems better than secretly meeting with her for dinner at that romantic Italian place by the Econo Lodge.
Whether we like it or not, sometimes past relationships come back. These days it is far too easy to communicate with an ex. (Facebook should just rename itself “The Ex File.”) Is it healthy to reopen old wounds? Perhaps it’s what he needs to assuage his guilt.
Try not to worry. Do you have any other proof that he’s cheating? He sounds like a decent guy with a conscious who needed some closure. Stop reading his texts. It’s unhealthy. Tell him that you don’t like that he went behind your back, but you trust him that their communication is over. I think it’s reasonable to ask him to not communicate with her anymore now that the bad blood has cleared. Do they want to be friends? That seems dicey if they had a rough break-up. Too much baggage. Now that he’s apologized and made peace, they should both move on and say goodbye for good.