@RDO, this man has betrayed you in the worst way. He has been lying and treating you with contempt, creating emotional distance between you to justify his cheating. He felt entitled to take a sledgehammer and shatter your marriage and sabotage his children’s well-being.
Find your anger. He needs to experience some sharp consequences. I would send him away for a while to let him know what the loss of you feels like. Tell him that YOU need time to consider the status of the marriage. If you allow him to stay right now, he will likely see it as a green light to cheat again and will lose respect for you.
He will be panicking that he got caught and promising you the moon, but, realistically, he was declaring love and making plans with OW only a few days ago. The attraction for her is still there. His weak character traits are still there. You cannot trust that it is over. He did not come to you and come clean, and on discovery he initially lied.
Most cheaters minimize on discovery, so you likely do not have the full truth. If feelings, sex and plans were involved, then the affair has probably been going on for longer than a month. If you decide to forgive, you must have the whole truth, so you will know what it is you’re forgiving. Everything, even what he told her about you and your marriage. Tell him that if you discover he’s still lying, it’s over.
His affair was not due to your being busy with children and neglecting the marriage. This is not about you at all. Do not allow him to deflect any blame to you. Please not do the Pick Me Dance to somehow compete with OW.
People who cheat have weak boundaries, a sense of entitlement, and poor coping mechanisms. The affair is a reflection of how they see themselves. Some are drawn to the risk, the unknown, the adventure, or to a previous version of themselves. Some use an affair partner as an escape or ego boost.
If you are going to try to work it out, I would insist that he do all the work necessary to help you heal and restore your trust. That includes individual counseling for him, to dig deep, investigate, and change the personality traits and behaviors that caused him to cheat and decimate his family.
He must provide full transparency with devices, phone records, bank and credit card statements.
He must take all your anger and hurt, and answer all questions whenever you ask.
You would benefit from counseling as a safe place to express your feelings and to organize your thoughts. Shine a light on his despicable behavior by telling people. You need all of the support you can get.
Bless you.