Dear L.,
Although I have not gone through this myself, I saw my Mom struggle with my cheating father. Since I am the only girl out of five, my Mom confided in me after I became an adult and explained to me what went on. She told me that she stayed up nights deciding what SHE wanted to do and what SHE wanted for herself, because regardless of the decision, her children (5 kids) would be OK. Once she decided what she wanted for herself, she made a plan to achieve that. She did not talk with my Dad to ask "do you have a girlfriend?". She says that that is a stupid thing to do because you already know. Your instincts are telling you what you need to know. So the only two things you need to decide are: do you stay with him, or do you go?
The success of deciding to stay and "work" through this will depend on your tolerance for distrust. Even after counseling, things are never the same afterwards and the pain and suspicions remain for a long time. Is it worth it for YOU? Can you really forgive? It sounds like you have already allowed a lot of his behavior to go on without consequences. When he has stayed out, do you let him back into your home and into your bed just like that? As long as you allow him to treat you this way, he will continue. You need to demand and get respect from him more than you need to catch him in the act.
For deciding to go, you need to think about whether you have the capacity to move on on your own. Do you have a strong support system that will help you in that tough time? Are you economically independent? Are you better off together than apart?
Once you make that decision, do everything possible to create the environment you want. Try not to get too emotional. Having a plan will help you take control of what you feel and act in a controlled and intelligent manner. This is very important so that you can maintain your composure and manage things with dignity, especially in front of your children.
My Mom did just that. She decided that she was going to stay together and that things were going to be on her terms. She completely managed the situation "behind the scenes" in order to make my Dad behave the way that she wanted. I tell you, it was a lot of work. No counseling or anything, just pure wit, strategy and determination. My parents have now been married for 46 years. They seem happy and looking back I think my Dad is thankful that she took matters into her own hands. No accusing, no playing victim, simply demanding the respect she deserved and placing strong consequences for his misbehaving did it for them.
I hope that you find the strength and wisdom to take matters into your own hands.
Good luck!!