aanons I do feel for you - my ex was - is - drink dependant, an alcoholic, call it what you want and I struggled for years to make him change. Finally I realised that he was never going to change or that even if he did I didn't want to be around to help him any more and we separated four years ago. I have no doubt that I made the right decision.
I do think your approach is understandable but wrong; imo, you have to accept that this is a problem that only he can change. Whether or not he does it with or without your support is for you to decide; but personally i would say that you need to be focusing your energy on your children and yourself.
It is very damaging for your dcs to grow up seeing you argue about drink and you will be so worn down, if you're not already, looking for bottles, checking how much is in them, trying to convince yourself that maybe you're overreacting, living with the repeated bitter disappointment each time you realise he's been drinking in secret again,etc etc. Why put yourself through that? and more importantly, why would you put your dcs through it? Because they won't be oblivious to it.
I don't understand alcoholism; I know that my exdp was really a good man but that his need for alcohol ultimately lost him his family life. I also know is that my family is so much happier now we are not dealing with it on a daily basis.
Have a look through the old threads - there are loads on here on this subject; there was a support thread when I was going through what you are. I remember Attila gave great advice, perhaps more thoughtful and considered than my "run for the hills " thoughts.
I really hope that it works out for you.