3rd I am really sorry you are in this mess.I know it was a stupid thing to do, as do you, but I kind of don't blame you, and I would not blame a man in similar circumstances. Despite the posters so sure they know how all mumsnetters think and post (and despite evidence to the contrary!).
I went off sex for a while, we did talk about it a bit. In the end I watched a programme about people whose marriages were breaking up and lack of sex was a reason for some. I realised if I carried on giving my dh the cold shoulder our marriage may be at risk. I even wondered if he may have an affair, despite the fact he is the least likely man in the world to have an affair!
Once we resumed a more active sex life I realised how much I liked it and that to some extent we had just fallen into no sex! (Or rather much reduced sex).
I think you need to discuss this and I kind of lean to the side saying you should be honest with him.
You do not mention children at all. That may mean there are none, or there are some and they are older/have left home, or are younger and are contributing to lack of sex as you and dh are both tired.
If there are no children, may I ask, if this is this by choice, mutual choice? I just wondered if this has played into your husband's thoughts at all. We had years of fertility treatment and lots of sex was to do with trying to have babies, which we had some success at, but not much! For a long time (for me) sex was about trying to make a baby, when we knew this would not happen I must admit it did limit the appeal of sex, for a time.
I think eventually you will need to learn to forgive yourself for this and I think telling your dh, and being honest, as you have here, may help you move on, either to a life with your dh, or with this other man, or maybe a new life flying solo.
Good luck, whatever anyone else thinks I can well imagine I could have done the same if I had felt my husband rejecting me sexually (rejecting intimacy with me) and I am not an especially sexual person, but it is (for me) intimacy which is key. I am also in that group of people who feels that this is not the worst thing you or he could have done. Rebuilding your life on a lie (that it did not happen) may end up being easier for him but very tricky for you, and that lie could, one day, undermine all your building work.
I have no experience of this situation and I hope you will find the answers you need.
The choice to tell must be your own as the consequences will be your own.