You were right to leave. Several members here who grew up in alcoholic households have said that its better to have no father than an alcoholic father. My children agree. I left three years ago and it's the best decision I've ever made. I should have done it earlier.
Maybe he's sober and going to AA and maybe he's lying. I don't know what your marriage is like underneath the alcoholism; in my case, it was dead years before I worked up the courage to leave. My AXH got into rehab and got sober when I left. I didn't want to be married to him anymore but I didn't want to prevent the kids from having a relationship with a recovering father. He stayed sober until he had secured shared custody, then he started drinking again and hell ensued.
In retrospect, I wish I had filed for full custody and conditioned his visitations on established sobriety, and custody to be revisited when he had at least 18 months of documented sobriety under his belt.
The kids will be fine. My youngest was eight when we divorced, and the first spontaneous comment she made about the divorce was "it's so strange, I'm not as afraid of everything now as I was when we lived with dad." That was after about three months.
You're doing all the right things. If you can get the kids into therapy, that might help - it's been a lifesaver for my kids.