Alcoholism is truly a family disease; it does not just affect the alcoholic. Your recovery from this will only begin when he is gone from your day to day life.
What did you learn about relationships when growing up?.
What is your definition of an alcoholic?. Some do act like your man, hold down jobs and drive. They do not all sit on park benches with paper bags.
This is basically a relationship that lurches and has lurched from one crisis to another; its never been stable for very long if at all. I doubt very much he will ever get around to marrying you either.
What do you get out of this relationship now?. What is in this for you?. If you are staying purely for the children you are basically showing them that on some level this is acceptable to you (and by turn they have to accept it as well). Any child will love a parent no matter how crap they actually are.
You are acting as many people do in such relationships with the alcoholic; as provoker and enabler. You also seem to act out co-dependent patterns in this relationship as well. Co-dependency often features heavily in such dysfunctional relationships.
I think life would be a lot calmer for you and these children if he was not in it day to day.
Is this really what you want to show them about relationships, that staying with an alcoholic is really the way forward?. You cannot stop him acting like a berk and you are not doing that either; what you are showing them is that this is acceptable to you.
He is likely self medicating any depression with drink but you forget here that alcohol is also a depressant. He is holding down a job, well for now anyway. He may well go onto lose that and everything and everyone around him and still continue to drink afterwards.
His father drank heavily and it is of no surprise whatsoever that his son, this man, is the same. Alcoholism can also be learnt.
He loves alcohol more than his children; they are certainly not at the top of his priority list. His next thoughts are centered around where the next drink is going to come from.
You and your children deserve a better life; they will also not thank you for staying with their dad should you choose to. They will despise you instead, perhaps even more than their own alcoholic father because they will be left with a raft of emotional problems pertaining to being a child of an alcoholic as a result.