The 3cs re alcoholism:-
You did not cause it
You cannot control it
You cannot cure it
His primary relationship is with drink and has been for many years. It is not with you or your children. He is still paying lip service to the problem and is not at all serious. He is still in denial and blames you for his problems. Such men do not change readily if ever and you cannot make that change happen for him.
He cannot be at all left alone with your children any longer.
He could go onto lose everything and still drink afterwards; that would be nothing to do with you either. If he does leave the marital home (and I would suggest you get him out asap) then where he goes is really not your concern either.
I also think your mental health would also markedly improve if you were to properly separate. His behaviour may well be the root cause of your own depressed state.
Life with an alcoholic is absolutely no life for you and your children, you are basically lurching from crisis to crisis. Its never stable and secure. He may also lose his job due to his drinking problem at some stage, that is not beyond the realms of possibility here.
Talk to Al-anon; start opening up to people properly. Alcoholism as well thrives on secrecy.
Aussiebean's comment is also spot on:-
"By staying and putting up with what he is doing, you are enabling".
Enabling only gives you a false sense of control and does not help him either. You can only help your own self here and that you can certainly do.
This initial comment of yours is very telling:-
"I don't want this, to be married to someone who sneakily buys booze and drinks until they're intoxicated. DC upstairs asleep but what if they'd been ill?"
You state that you are afraid of being on your own but you're pretty much on your own now within this marriage. You are married to a drunkard. Staying within it will simply have you in a worse situation down the line and your children will also be far more aware of their drunkard for a dad (leaving them with a raft full of emotional problems linked to their dad's alcoholism).