Well, just don't, "let it go," if it is weighing on your mind. It's important to apologize. Once I apologize, my side is clean. Then I go on to not do that again, which reinforces the amends. You don't have to mention your problem with alcohol until they are old enough to understand - but they likely remember the "scene." Knowing that it isn't going to happen again will be very comforting to them.
Apologies for bad behavior are always welcome - not only that but when your kids say, "That's okay, momma," you'll get a huge relief. Or even if they cry or whatever, it is honest healing taking place. They will love you always, but keeping your side of the street clean is important.
I had to do my forgiving without the apology from Mom. It isn't the same.