unyummy
The only person you can help here is yourself along with your children. Not him.
You cannot scare him into doing anything re his drink problem. The 3cs re alcoholism apply - you cannot cure this or control this. You also did not cause this to arise.
Your man has no intention at all of giving up booze; he is quite happy to continue as he is. He was only able to stop drinking for a few days; also you only have his word for it that he has signed up for counselling and addiction issues. Talk is cheap, its actions and long term actions at that that matter. He told you what you wanted to hear from him. They are truly masters at manipulation of others and denial.
BTW as well make your house an alcohol free zone. If you do happen to drink alcohol with him stop this behaviour, this is only enabling him more.
When it comes to alcoholism the whole family need help and support, not just the alcoholic.
I was not at all surprised to subsequently read in your post that he had recommenced drinking alcohol; he has no intention at all of wanting to seek proper long term help for his drink problem. All that he said to you means nothing; actions in the long term speak far louder than words. You are his enabler now, you enable him by being there to clear up after him. Stop asking him too if he's had a drink, you will know anyway.
Perhaps you should kick him out as you write. You don't yourself have to move out, why should you when he is the one who has caused all this to arise in the first place.
Look at this problem another way too.
What are you both teaching your own children about relationships here?. Growing up in a household where one parent is an alcoholic brings the children their own set of problems. Your 4 year old is no doubt already aware that something is amiss but cannot put it into words. It will affect them as children markedly. This is NOT a legacy you want to leave them believe me.
The following is also going to be hard for you to have acceptance of:-
There are no guarantees here, he could lose everything and he could still drink. You cannot allow yourselves to be dragged down with him any more. You can save your own selves but you cannot rescue and or save someone who does not want to be saved. He has to want to help his own self, enabling him as you are doing will only prolong the agonies.
If you set an ultimatum you can only do so once and you have to follow it through. Making them repeatedly loses their power.
You need to talk to Al-anon, you need real life support too.