I would think that if you have a written agreement with him, his consent to your move, you should be able to move out of state with the kids. That’s the law where I live. Must have written consent.
If moving is the only way you can provide for your kids, I don’t see any choice. If you stay where you are, it seems that you would now have to rely upon someone who has proven to be unreliable, regardless of the reasonn why he's been unreliable.
As already mentioned, there are lots of people living apart, even with kids, for economic reasons. And there have always been families that have done that. To me personally, this wouldn't really be about alcoholism or recovery, divorce or legal separation as much as about financial necessity. 35 years ago, my parents lived in different cities for 10+ years for that reason.