ChristyC,
You need to redefine your own definitions of an alcoholic because the image of an unshaven man drinking from a brown paper bag is not just an out of date image, its just plain wrong.
"I know where I am at now - he has a problem that he needs to address, but now I feel a failure if I don't stick around to help him through it".
Re this comment of yours you are right with the first part of this sentence but completely and utterly dead wrong with the second. You will not be a "failure" if you don't stick around - you need to realise that you cannot help him. His Mum is right -you have enabled him to date and it has not helped. You have instead all been dragged down with him. Him being sent to a Crown Court must have been bloody awful for you as a family - you must have had a whole range of emotions not least of all embarrassment on your family unit.
He needs to reach his own rock bottom and even then he may not stop drinking. There are no guarantees. He is responsible for his own actions, not you. These are not your issues to address
If you really want to help him he must seek help on his own and have no contact whatsoever with you.
"He says that I am throwing everything away just because of how he behaves when he's drunk but I say he already did that by behaving badly when he is".
You are correct. It is typical of such people to project blame onto everyone else around them. You cannot also reason with such people. Alcoholics are by their very nature selfish.
If you are answerable to anyone it will be your children. They won't thank you for doing nothing if you ultimately choose to stay with such a man. They could well accuse you of putting him before them.
You need to seriously consider what you are teaching them about relationships - it is perfectly okay to leave an unhealthy one. You will do them a huge favour by teaching them that it is not okay for Mummy to be treated like this.
Do call Al-anon; they can help you if you yourself want helping.