Mom, I think that you should write the letter. But I don't think you should mail it to him while you are with the OM--at least not yet. If he re-marries, then it's a different story. You can mail it to him then. After its clear that you/him are no longer an "option" But tell him that you don't expect or ask for a response.
I have received text messages from my H several times telling me how sorry he is for hurting me. But I have no idea what to do with them. He is in the midst of what looks to me like an addictive/insane affair, he has hurt me terribly and our children terribly, and I really don't know what to do with his guilt. While I think it is healthy for him to feel sorry, I feel like he wants me to "carry" that for him--to tell him that it was OK. But it wasn't OK. He had forgotten all about the "love, honor and cherish" part of marriage, and had for many years, and I'm still in a lot of pain from that. To ask for me to console him on top of that seems selfish to me. I think it is really good that you feel sorry-- you are growing and he is growing, but asking for forgiveness from him could actually place an added burden on him. You need to carry this yourself for awhile, and the other posters are right--first, conduct yourself with compassion and caring at all times, then, when its clear that you have both moved on, you can mail him the letter. Don't ask for or expect a reply. If you are kind to him and kind to your kids, eventually you may earn forgiveness from him. The risk right now is that it will feel disingenious to him if you are still with the OM, and especially so if you are feeling insecure about that relationship...it could be an unintended ploy by you to keep him under your influence just in case things don't work out with BF/OM. So, write the letter, but give it more time. Make sure you truly know your motives, and make sure he will genuinely understand your motives. Neither of you probably knows what the future holds for certain at this point.