My (26F) boyfriend (26M) is a "recovering" alcoholic and I'm having trouble coping

I (26F) and my boyfriend (26M) have been in a relationship for over 2 years. We've known each other around 6.

When we first got together, he regularly drank platinum vodka -- talking around 3 of the 1.75L bottles a week. That was just at the house. We went out pretty often also.

I wasn't aware of how bad it actually was until I started staying at his place. He'd buy 2 bottles at a time and mix that with whatever can of pop he had on hand. The bottles started to pile up. I started to get concerned. And then I started finding them stashed, empty, in random places in his room. In the closet, under the bed, in shoe boxes, stuffed in a shoe rack, etc etc. one day I put all of the bottles I found in a garbage bag and planned to say something when he got off work. More just to prove a point, not to nag him. This was still less than a year into our relationship.
When he did get home and saw the garbage bag, I tried to talk to him, but his response was: "yeah yeah yeah, if it kills me, it kills me."
That hurt so bad to hear. I cried. I still think about it.

Fast forward some time later, I told him how his comment hurt me and that I was just worried about him. Some time after that we had agreed on only having a drink on the weekends. Okay.
But I was still finding bottles, and on our one year (a Tuesday) I come home from a closing shift and he's plastered. Couldn't even stand. I left that night and stayed with my brother.

Fast forward some more time later, there were more hiccups, and he had agreed to cut out vodka. He'd have a few beers on the weekends. Great progress. And he actually followed through. No more vodka. Even as of current.

Until some things happened and I would find bottles of fireball. I'd dump them every time and call him out on it. Always "I'm sorry, it won't happen again." Or times where he'd lie about how long something has been somewhere. Most times I found it all on accident. I never want/ed to nag him. I just wanted him to be honest and open with me. Communicate. There'd be periods of time where i genuinely thought everything was going great and that he was doing good. Sometimes he was, sometimes he was lying. It hurt, but I knew he had made progress and I didn't want to give up on him. I know alcohol abuse is a big issue. And it runs in his family. That's another big reason why I so badly want to help him still.

Throughout this whole time I had talked to one of his friends about it, and he's been really kind and supportive. My boyfriend knows I've talked to his friend regarding the issue. Maybe I shouldn't have, but I wanted his parents to be an absolute last resort.

We recently moved into our own apartment together, and before we signed the lease, I told him no hard liquor in the apartment, and that he better not let me find anything. I told him I didn't want to be trapped in a lease if he were going to keep lying and hiding things, because it was really taking a toll on me mentally. I was paranoid all the time. My trust was broken. I never believed a word he said, even when he was being honest. I told him there's no relationship if there's no honesty or communication. He agreed. Whatever.

Since we've moved in (as of the beginning of July) I found seven bottles of fireball. All on accident. I don't know why I keep sticking around. It's messed me up so badly. I feel like I have to look for it now. He games in the spare room, and I feel like I need to pop in every five minutes just to check. I feel like if he's in the bathroom then he's drinking. He could deny all of this and I would not believe him. I told him the other day that I'm seriously questioning our relationship. He says he's sorry, that it won't happen again, he doesn't know why he did it, that he loves me, that he's trying and he'll do better. But that all goes in one ear and out the other at this point.

So my question is:

Is this something we can still work through? Or are we just better off breaking up? I desperately need advice.