Don't wait too long to confront. If she hasn't met him in person, that is coming soon.
When you confront, tell her you want to improve the marriage and ask her to commit to that. You love her and want a good marriage. You want her to be happy and you are fixing your faults.
If she still wants to be with the other man after that, tell her that the other man has had a huge advantage, he has been on his best behavior, he has been fighting against you, but you have not been able to fight back, so both him and you have been comparing to you, but you have been unaware. All of the relationship with the other man is not reality-based - no kids, no chores, no finances, no aging parents or extended relatives - for her and other man, it has just been sexting and I love you's, all passion and no reality, as opposed to you, with three kids, you are starved for passion for a good reason - lack of time - working full time, dealing with kids, chores, finances, etc.
Tell her if she wants to be with other man, after you speak your mind, then tell her to go be with him. Ask her for his phone number, tell her you will call him yourself and let him know she's coming, bag and baggage, and you are going to move on and grieve your loss, and then eventually hopefully find another woman who will not cheat and lie secretly behind your back. You, too, will find a new great love. A win-win. Good bye and good riddance. The kids stay here with you. Her actions cannot be undone. If she comes crawling back later, maybe you will take her back, maybe you won't, you can't say. You thought cheating would be a dealbreaker but it wasn't, and you think her leaving to be with him, refusing to end contact will be a dealbreaker, but you will take it a day at a time.
Tell her you will move forward, though, and you will respect yourself as a husband and as a father and as a man.
Now think about what you want. Do you want a wife who grudgingly stays with you out obligation or fear of unknown with other man?
Also, though, if you love your wife, treat her like an adult. And if she is acting childish, help her see what she is doing and where she stands. Grown adults can be gullible, can be wishful thinkers; grown adults purchase pills on informercials in the middle of the night thinking taking a pill will make them lose weight, or if they buy this real estate course, they will become millionaires. Because cheating is such a predictable behavior, and I have read it for so long, I can tell you that your wife is like those people.
Is the other man married? Is he a real person? Does he have a job? How much does she really know about him? I, and many others here, have seen that other men lie. Over sex. Would you believe that a guy would cheat and manipulate to get sex? Apparently there are quite of few cheating wives who don't know that. Or think they are little special unique snowflakes, so it is not happening to them.
And on that point, there also are many betrayed spouses who also think they are special little snowflakes, that their cheaters are better than those other cheaters who are populated in the other threads here. Yes, the cheater in that other thread is a terrible person, but my little cheater, mine is really a great person!!! I can tell you from my point of view, I see no evidence that your wife is any different than mine or any other cheaters here. Your wife maybe didn't have sex, not physical YET, but she would if she had the opportunity. Just luck. There is a lot of luck involved in this. A lot is beyond outside your control. You can control yourself. If you're trying to control your wife, step aside, you're down the wrong path.
As you look at this whole shit sandwich you are starting to take a bite of, know a few truisms. Watch her actions, not so much her words. If the words and actions conflict, believe the actions and ignore the words. If she says she wants to be with you and work on the marriage, but then she stays in contact with him, then believe that action.
She will minimize. She will flat out lie. She will blame you. She will rationalize. It really is extremely simple. You both vowed, she broke them. Does she want to make amends?
What should you ask for?
1, she ends contact from him now and forever;
1a, she makes an effort to show you that she is ending the contact, she gives you evidence she has done it, either lets you hear it, or see the texts, she offers up all her accounts and devices, passwords, and says you can look whenever you want because she no longer has any secrets. (Should spouses have secrets between each other? What is the difference between privacy and secrecy? If there is privacy between spouses, what should it be? Would privacy be in an email or text or social media?) She blocks him, changes her phone number, deletes his contacts, terminates her social media accounts - these are other things she can show that she truly wants to end contact with other man.
2, she tells you the full truth. She shows you information about other man, answers your questions, and the story she tells makes sense.
3, she engages back in the marriage (as do you).
The number one sign of cheating is hiding/guarding the phone. The number two sign is the cheater's behavior, everything you say or do annoys her, she is short with you, distant with you, cold with you, not affectionate. Once caught, it is easy for the cheater to get around and stop hiding/guarding the phone and still cheat, but changing the behavior is difficult for any length of time.
Is the other man married?
Who knows besides you and her and other man? Any of her friends or families know?