Holy COW is there SERIOUSLY a law saying spouses MUST PAY 1/2 of bills?!!? OMG!!! I dint' know the law specified the shares any roommates had to pay for rent much less spouses! I'm BLOWN AWAY.
OK, Anyway, here is what I would do, and I'm SO SORRY you are going through this. It sounds like you want to leave but feel you can't because of the lease. So I'm going with the assumption that you are willing to leave him in general since he drinks and treats you terribly and is not being a good provider-these things will not change. Therefore you should leave, and it sounds like you would.
I would not try to negotiate in any way with him on changing anything he does, since that will not happen. You need to let him know you cannot live this way anymore and you want to separate. You know the lease states you must continue to pay half of the lease. (I'm still blown away-are you SURE?-check on this law and hopefully other moms here have input on that law too-I would have a free consultation with a divorce lawyer and ask about this, they may know a way around it). Tell him he needs to find a roommate to move in after a specific date.
Before you do this, take the time you need to decide when you would or could move-I know it's very hard if you're not able to save your money because of this insane law, but think of all your options and work toward this goal. Worst case scenario you'll have to wait out the 7 months, but you can still be planning for your future without him, and get a different roommate when the lease is up.
I know it seems like you'd be no better off financially with a different roommate and the expense of a move-in deposit etc, but you'll be freed from him and able to live life from a whole different angle (and meet someone else) that will lead to a new future, unlike if you continue to stay with him. It may be good to move in with friends or family for a bit if you have that option.
If your relationship was great, and you had agreed to split everything, you would have spent that much money anyway for the next 7 months, so don't think of it as a financial loss that will permanently ruin you if you have to wait it out and pay. Time passes quickly, and it gives you a chance to plan with a roof over your head for a while. Just making the decision to leave will free your mind a bit rather than feeling overwhelmed by the grim future of living with a man like this forever-which makes every little hardship seem debilitating. Keep things separate in your mind and make a plan you are pursuing. This will help with your depression, because you won't be dwelling on the frustration of living with him. People can't always act right away, but you will be able to change this. Be patient with yourself and focus on what you need.
You will come back stronger in the future when this is over and this lesson will help you. As in, never settle for this treatment ever again-you work too hard! It is so common for strong women with self made resources to end up with men who won't support women. Start over! There is a difference between two people pitching in for a greater good and one person hoarding and and treating a spouse like a roommate-which you already know. I really pray for you to get through this to the other side. Best wishes!
ps, I just want to say, that if you are really taking action to leave because of his selfish hoarding, there is a tiny chance he might try to change if he wants to save the marriage-the separation may reveal this later-but do NOT take that upon yourself in the form of trying to make him change with the threat of leaving, or having counselors tell him he really should not act that way. This will only waste more time and divert your focus from what is best for you and the kids. As long as he can act this way and keep his family, he will. You must take action.