I am really sorry! Do you have a supportive family or somewhere to go? Addiction is difficult. He may care more than you realize because the nature of addiction is that he is ill. They 'can't' stop until they address the addiction. Your leaving may be his wake up call. I don't think you need to be with anyone who has been having sex outside your marriage but your kids will need to have a relationship with him and be with him so hopefully he will get some help.
We're here to support you and again, sorry you are going through this! hugs
Sweetheart, it is time to find a good and really mean lawyer. Call your local legal services clinic if you don't have any money. To find a legal services clinic, contact the nearest law school or look in the yellow pages. You can worry about the psychology of having been betrayed and how bad it feels later, right now it is time to nail his behind to the wall.