Hi, and welcome!
FWIW, my first husband started off almost exactly like your boyfriend. He went to one meeting, decided he wanted to try to do it on his own, and after some more disappointments and taking a break from the relationship for a bit, he went back to AA on his own, quit drinking almost immediately, and has not picked up a drink in the past 31 years.
My point being, there is hope. My suggestion for you is that you learn all you can about alcoholism (lots of good info on this site), try some Al-Anon meetings, and maybe even go check out a couple of AA meetings on your own. It sounds as if he is making some moves in the right direction, but it may take awhile for him to get there. Of course, some alcoholics never do.
Meantime, you need to be taking care of YOURSELF, which is the only person you can really change.