I'm very sorry for your pain, kjlk. I imagine it must have been the most horrendous shock for you.
I would agree that your husband has neatly compartmentalised his secondary life for the four years this has been going on. I would also hazard a guess that the 'once a month' you think it has been is somewhat more than that. A woman wouldn't, I think, make plans for a future with a man that she only sees once a month.
That also begs the question as to whether your husband has feelings for her. I don't believe that he can be with somebody for four years and no develop deep feelings, feelings on both sides. Otherwise, she would be one of a few/many. I imagine he's trying to spare you (and him) from the pain this realisation will cause you but I don't believe it was just sex. It would be just sex on both sides then and the note wouldn't have happened.
I don't think you should tell your children right now. They're adults and they will cope BUT that bell cannot be un-rung. This is not about them they have nothing to do with your relationship as husband and wife. Your marriage is between you and your husband. His relationship and your relationship with your children are and can be, two distinct things. It would be up to him if he chooses to be a dad/granddad should you decide to split up. His relationship with them wouldn't have an impact on you and your love for them. They would be torn between you and, if you decide to stay together then that pain is for nothing really. Spare them or tell them - you have the right to decide for yourself. Will your husband tell them do you think? Probably not if he thinks he can keep it quiet.
What are the most pressing things for you? Do you feel that you want to talk with your husband? Do you think there's anything that you could see past/forgive him for? Do you know what HE wants? I know he's not the important one here but what he wants might be a starting point for you to decide what you want to do.
I don't think you know everything, not by a long shot. I think you have the bare minimum, stated in panic and now he's feverishly planning scenarios based on what he thinks you will/won't put up with/find out. If you're going to leave him then I would go and see a solicitor to find out what you are entitled to. That may help your decision.
If you decide you want to make a go of your marriage and stay then you have a decision to make - blot out what's happened, demand absolute faithfulness from this point on/agree to an 'open' marriage so long as you're not having your nose rubbed in it - or you can grit your teeth and demand honesty so that you can come to terms and keep going with your marriage.
You get to make the decisions at the moment, you don't have to be dependent on this man at all.
I'm so sorry.