Firstly, I am really sorry to hear you are going through this.
I agree with @ScreamingLadySutch, you cannot change him, he is alcoholic and his relationship is with alcohol; nothing else matters. Occasionally he might tell you everything you want to hear but mostly he will show you by his actions that you and your children are not his priority.
So, focus on yourself. Because of your living situation (council accommodation) you must move his belongings out of your bedroom. Do you have a spare room? If not, set up his things outside of your marital bedroom, put them in the shed or garage and put a lock on your bedroom door, the type with a key. You are separating yourself from him.
Do not have any further sexual relations with him.
Do not buy food for him.
Do not pay for his phone.
Do not do his laundry.
He is on his own now and you are living separately.
If he tries to save your relationship by telling you he is in AA that's fine and good for him, but don't allow a reconciliation between you until he has been sober for 3 months. You can purchase alcohol testing kits from pharmacies, if he is telling you the truth he won't mind testing.
But a word of caution - alcoholics lie, they lie and they lie and they lie! They gaslight their spouses and you will constantly be watching him for any signs of alcohol abuse. He will put you on the wrong foot by making you question yourself when you confront him because you know he's still drinking.
You might also start the madness of looking for evidence of his drinking; bottles, open or empty, which you will come across from time to time - after he has sworn blind, on your life, his life and his kids lives that he has stopped, remember alcoholics lie. You will expose him and he will make up excuses and you're back to square one.
Please don't do this to yourself, if you don't want to live the life I've explained above, then separate yourself from him.
Do all the things you need to do: learn to drive, talk to CAB, get a solicitor and go to Al Anon but for your own sanity, get him out of your life.