I, just like the others stated am not a professional therapist,but I would like to say a few words of advice or suggestions...
First thing is you are very brave to get on with that story. I mean that in a good way. The first step to anything you want to fix, is to come to terms and admit something is wrong. It looks like you have started that. At least to yourself. No matter what, you have to look at the fact that you are not just telling this story to hear other people opinions. You have also told us this story because you may be starting to realize that truth and honesty with yourself is KEY when attempting to stop drinking. It might be that these other women keep your mind off the reality of life. I don't mean about you as a father(sounds like you love your kids to death), but you as a human being in general. It is natural for us to want to feel needed and we ourselves have to learn how to balance it properly. I am on DAY 8 without a drink and already feel confused and frustrated. It seems like with or without the other women, you are not happy with your wife. and have not been for a while...
You need to find happiness within yourself that has nothing to do with your wife, kids, or the other woman. Spend a couple hours alone and maybe think or write down what you are feeling at that moment. Then expand and write about reasons why you made these choices..then think about what needs to happen for ANYTHING to change. That includes cheating, drinking, fighting with the wife...I only say this because you did ask for some advice, and no one has the right to harsh on you when you come to this forum. It seems smart for anyone to either suggest no advice(like other posts) to give for the relationship with the wife, but can give advice on the relationship with drinking. Consuming the amounts you can in a day will definitely cause tremors and unsteady hands.
Do you go to counseling right now. I would suggest it, as well as being totally honest with your therapist. You need to talk to people about this and not let it continue. For your own sanity as well as the guilt you may feel later on down the line. Sounds like your kids love you, so think of yourself as their hero. How would they feel if they knew what Daddy did when he was with mommy? I say this with kindness because it can be a helpful tool when trying to organize your thought. Always remember you have little youngsters that look up to you and think you are the world. Show your kids that you are and can be the world. You have to give yourself strength and reason to stop. You had the strength to post this..i believe deep within you have answers to many questions as to why you have been living your life like this.
I hope you don't mind my words..I will not hang you for your post, but I would love to hear that you are trying to change the situation. Good Luck..