First of all. let me say that I am sorry this happened to you. This is a long time, 23 years, and to experience this type of betrayal is not an easy thing to get past emotionally. But it can be accomplished, and I want to encourage you and give you some general best practices that you can use during this difficult time.
I feel that if he wanted to leave you, he would have. Clearly there is something still in evidence with the relationship emotionally, and his willingness to be intimate with you is also a sign of this. Now that he is found out it is normal that there would be a period of time where he would be emotionally distant.
But, he also has a great deal of responsibility to own up to. He cheated on you. He obviously distorted the truth with you and he betrayed your trust. And he is not taking responsibility for what he did; rather, he seems to be acting distant and aloof, which is what someone who was hurt might do...yet he did the hurting!
The other woman: It is best, XXXXX XXXXX consistent in counseling advice, that you do not have contact with her. She is a problem yes, but she is his problem.
If he wants you and the children he needs to cut off all contact with her. And, your interaction with her places responsibility for his actions onto you, and that is not fair to you. Besides, she seems to be lying about a number of things and you do not need to be any more confused and hurt than you already are. Talking with her will only make you feel worse and she is baiting you with her writing, hoping to draw you in on her terms. So, do not play that game. Ignore her instead. If you feel you still need to confront her, save that for when things stabilize and you know where you stand. Right now is not that time.
So: A few needed things.
I would highly suggest that you see a couples counselor. If he refuses to go, you go yourself for you need the support. Many women also make participation in counseling a requirement of any partner reconciliation. He may go or not, but at first I would like to see you get some support and perspective. You need someone face to face who can help you.
He needs to cut off all contact with her. If he does not, he has shown that he no longer wants his relationship with you and the children. He must cut her off, totally. He cannot have both, ever.
He, not you, needs to show some behaviors of reconciliation. He is the one who messed this relationship up, and I would back away and let him make amends, for you trying to be with him is not the issue. He is the one who must show the remorse and make the effort.
I would make an effort to get support for yourself, and be clear in my demands of what he needs to do to retain your relationship. I would not go near the other woman lest you give her a chance to try to hurt you more than she has. And I would not hesitate to know that this was not your fault and that he must now show that he wants this relationship to work. Steven