Hi

Mate the very last thing you need to do is to keep running your mouth and telling more people. You made a mistake proactively bringing this up with HR in the first place. If it’s just a public drunkenness and you didn’t do anything particularly dickish/aggravated then it most likely won’t be a big deal and everything will be fine. If and only if worst case scenario does happen, then yeah tell your connections on the team and see if they can vouch.
 

But you should have known or researched whether this type of thing even shows up on background checks before mentioning it. There is absolutely no need to volunteer negative information about yourself to an employer that no one explicitly asked for that gives them a reason second guess you. You don’t get brownie points for tattling on yourself for something minor. If it comes up in a BG check and HR confronts you then it is what it is and you need to be humble about it and tell them what they want to hear. But there is little benefit to in trying to preempt that.