An addict who blames their partner for their addiction is not ready to face up to it imo. Addicts have to be honest with themselves first and foremost, and take ownership of the problem.
(Disclaimer: this may not apply where the addict is self-medicating because of untreated/undiagnosed MH issues).
All the time your husband is treating you as part of the problem, he's avoiding the problem. The counselling is just a smokescreen, so he can be seen to be doing something, even though that "something" may be pointless. Don't let him turn this into an "us" issue when it should be a "me" issue.
Not only is he avoiding facing up to the alcohol issue in any meaningful way, he is jeopardising the welfare of his children. Claiming UC for the family and failing to pay the rent is financial abuse: that is family money, to keep a roof over the family's head.
As you are in arrears, it should be possible to ask UC to pay the rent money direct to the landlord (this is known as an "alternative payment arrangement). However, you are currently not living together.
This is a change of circumstances that must be reported to DWP and I advise you to do that and start your own UC claim. Not only is that the right thing to do in terms of benefit law, it is the right thing to do for the security of your family. And it will put you in control.
Please bear in mind that if you were to lose your home because of rent arrears, the council may well refuse to help you because you (the collective you) have made yourselves homeless intentionally by getting into rent arrears. Of course, you would be able to challenge this on the grounds that you were being financially abused, but the outcome of such an appeal is far from guaranteed, and it's hassle you don't need.
I'm sorry if this sounds harsh, but it is exactly the advice I would be giving if you were a client of mine, and I think you need to know what the implications of carrying on as you are could be.