Run Away. Fast. And now.
This man has been lying to you all along. Look back and see. You are a co-dependent, like most of us. He says and does exactly what he needs to keep you hanging on. But it is all about him.
20+ years into a relationship with an alcoholic and I wish I would have left at the first hint. I had nothing as definitive as you have posted. My parents voiced some concern, but they didn't come out and say it. As an only child, I assumed more of the "no woman would be good enough" for my mom theory.
Then, after being married for a few months, I started to see it. As I started working an exit plan, she turned up pregnant. And here I am. $100K poorer with nothing to show, few friends, stifled career, and 2 kids who have grown up with an alcoholic parent.
It is "possible" he will mend his ways. And people like us tend to always look at issues from a glass is almost full point of view. However, the "probability" of him recovering is very low. Your odds of winning the powerball lottery are higher.
You asked what to do. I say run away. There is no way I would ever wish what I've gone through on anyone. If you don't run away now, you will look back in 20 years and wonder why you didn't.