Firstly, I’m sorry about your situation OP, I feel for you. Remain strong and objective, and you will thrive through this whether you choose to save your marriage or choose to move on.
This message represents my opinion based on a lifetime of experience from my perspective. It doe not mean I am right/wrong. Just take from it what you feel resonates with you and ignore what you feel doesn’t. That also goes for anyone else who reads this.
Yes, your DH is all sorts of wrong for being selfish/brain dead for choosing sex over his DC and your marriage… if YOU decide to work on your marriage with him, he needs to make it up to you over a long period of time, demonstrating his devotion to the marriage and DC. However, you also need to work on this as the issue was created by both of you over a prolonged period of taking each other for granted.
The mistake many couples make is they do not put each other first once they settle down and DC tend to become absolute top priority, leaving the foundation of the relationship (the cornerstone, if you will) playing bottom fiddle, over time allowing it to crumble. Yes, it would be great if all partners could just remain happy no matter what, but we are emotional beings by nature and we cannot help how we feel. Putting partners 2nd in any way is what leads to affairs. DC can still be devoted to lovingly without any neglect, but it was your relationship with DH that created the family in the first place. You neglected each other over time and unfortunately the result is the situation you are currently in.
Only you, OP, will know what you truly want. If you choose to end the marriage it still is best to try and maintain an objective dialogue to minimise the mess it will leave behind. If there is little equity in your property, there is no need to ‘buy him out’ as he is equally responsible for a safe roof to be over his children’s heads. This does not mean taking as much equity as he can and setting up a new relationship elsewhere… He needs to take the financial responsibility/repercussions of his actions. If he is not taking DC with him he needs to stand on his own two feet with his salary, whilst supporting his DC financially too. The key thing here is a robust agreement to support DC amicably without any animosity and stress to the DC.
I sense you would prefer to save your marriage. If this is the case then you both must go back to basics and communicate more than you ever have, exploring how you both feel and leaving no stone unturned when discussing your feelings. Not just now, but forever. You may choose to seek professional counselling (which would hit home with your DH the mess that has been created). You can explain how your DH has made you feel and he can explain what led to him choosing to do what he did and how that made him feel. If you can achieve this level of honesty and save your marriage, it will also elevate your marriage to a new level and will possibly feel like a brand new and exciting relationship. I must stress though, that if a single detail is not explored and not discussed and is swept under the carpet, then you risk the cycle you are currently in beginning again. In these situations, it is the unspoken word that is the most damaging. BTW, sex is also significantly enhanced when a couple reaches a certain level of emotional connection with honesty at the foundation. Most of society has never experienced what I am talking about it. You will know if you have.
I personally would not get hung up or obsess over what word your DH said to the OW as it needs putting into context. At the time, he was trying to get laid and I have learned that most men would say anything to achieve it (I am a man, but not most men). The key point I see here is that he did not leave and he still remained with you, so that is a good sign and something you can build on if you choose to go down the path of complete honesty and transparency whilst rebuilding your marriage. The OW also felt the relationship was wrong, so she pulled back. This tells my that they both have at least some kind of moral compass but they were too weak to avoid crossing the line in the first place. Your DH is begging you not to divorce but at present he has no idea the level of work that needs to be put in to remedy your marriage and take it to the next level. He will need your guidance and be willing to work with you every step of the way no matter how much he is out of his comfort zone.
Finally, the deep emotional connection you feel they have/had. I fully appreciate this hurts, probably the most, but you will be surprised how easy it is for humans to “find their soulmate’ after just sharing one piece of personal information followed by sex. We can be pathetic at times, but in reality we extremely rarely do find a true soulmate and it is often a false reality we have created in our own fantasy. I’m not saying your DH/OW are soulmates by any means, just making a point. This affair would have begun with one of them simply telling the other they are not ‘getting any’ at home, the other says ‘me too!’ and all of a sudden the barriers are down and they empathise with each others situation, share more personal problems, an ‘emotional’ bond is created… and an affair starts - do you see that an affair also starts with honesty/communication between two people. Your DH just found this with the OW unfortunately. Honesty/communication really is that powerful, so use it to its full potential if you choose to work on your marriage. The truth is, if DH was with this OW with 3 children in the same house, this same pattern would likely repeat over time. This is why people who have affairs are often weak willed and emotionally immature. Saying that, you often need to experience a situation like the one that is happening now in order to improve yourself and enhance your state of being.
A different perspective that may help you feel better is that if you choose to work on your marriage, this next stage of your relationship could be better than the first - an exciting prospect! If you begin this route, start by showing DH this Mumsnet thread and every single post. He needs a reality check and to see how many posters said ‘dump his ass’ or have given their opinion on him. That may (or may not) hit him harder than the emails you discovered did to you. Either way, it is the beginning of communication and honesty.
Good luck OP, whatever YOU decide!