Well, it sounds to me like there were a lot of underlying issues that suddenly were brought to light. Without knowing exactly what those are, it's really hard to determine how you can help save the relationship.
However, there were a few things that you mentioned that were a bit surprising: First, you disparaged yourself a lot in that post. I've found that when I apologize, it doesn't help to go "I'm an awful person." Mainly because it puts the other person in such a bad position. They can agree with you (making you feel worse), or they can console you, which doesn't actually solve the problem. Maybe next time, you can say "You know, I am sorry I said a bunch of things that had no right to be said when I was drunk. Obviously, these are issues that I've been neglecting to talk about, and I should have brought them up at a more appropriate time. I'm sorry that I said it, and I hope you forgive me." and leave it at that.
When he said "stop saying things to make me feel better", he most likely felt you were being disingenuous. You know how if someone says something over and over and over that you get tired of it? It sounds like that's what happened. You felt incredibly bad, apologized, and he forgave you. Yet, you continued to apologize. I understand your mindset. You feel you messed it up and you feel you should make it better. However, if you apologize too much, it tends to give off the vibe that you don't necessarily mean it (even though you do.)
He said that he forgave you for the incident. He also said he wasn't breaking up with you. These are both positive things. Remember, though: One conversation didn't get you into this mess. There was a lot of built up and neglected emotions that led to this point. One conversation won't fix things for you, either. You, alone, cannot save this relationship. Relationships take two people. I'm sure there are things that you need to fix. I'm sure there are things that he needs to fix. However, relationships only work if two people work together and have the same goal in mind. He mentioned he isn't putting as much effort in anymore, and that's something you should strongly consider. If he isn't willing to put forth the effort to make the relationship work, he isn't the person to build a relationship with.
TL;DR- It takes two people to make a relationship work. Both of you need to do some work. One thing for you to work on: Give one, heartfelt apology and let it be. If they forgive you, fantastic. If they don't, or need some time, give them that, too.
Best of luck!