No you can't have a functioning relationship with an alcoholic. As colditz said, he is already married (to booze, the whole process of being an addict). Alcoholism - or any addiction - is extremely toxic to personal relationships.
alarmed tbh at Heroine's suggestions: addicts usually sniff out an enabler and to drop everything to pander to his addiction is what enablers do: they keep it going and are part of the addiction. Addicts don't have the luxury of becoming "less active": they have to STOP, there is no middle ground. You also can't 'make' an addict stop - anyone who thinks they can is deluded: an addict has to want to stop and tbh they don't care one bit how much anyone around them is suffering because of their addiction, just as long as they get their fix.
I don't envy you isiteverpossible - this is a very hard situation to be in. But you are nothing like the first and will not be the last. AlAnon is the first port of call tbh - keep looking until you find a group that suits you.
It is very possible that the school have already reported the situation to social services (I'm so sorry) and you need to act quickly to get in first. Social Services protect the chilren's needs first and foremost and an addict in the home is a disaster. Contact womens aid who will give you some excellent advice about where to go from here. He has to leave, not you and the kids. Also try the Alcohol Advisory Service - I think that is a national org?
Please don't minimise this problem (oh well, he's down) because it is very serious, apart from the obvious danger of driving the kids around whilst drunk
for example. ok he's lost his job and that's hard - but many lose their job but don't develop a full-blown addiction. ie it's a reason not an excuse.
and btw you haven't 'got yourself' into this situation: he has.
good luck, be brave. think "single parent" and carve out a solution. There is a LOT of help out there. Look after you and the kids - leave him to his 'wife' who doesn't complain or make things difficult for him (I am not being bitter, just tough on the addiction)