Most of the people that I treat for Substance Use Disorders have said they were “sorry” like a bazillion times, so a regular apology isn’t gonna cut it.
You gotta get away from the generic “I’m Sorry” and for heaven’s sake DO NOT say something like “It’s all in the past, can’t we just move forward?”!!!!
It pretty much sucks to be reminded of all your past wrong doings, which is why you need to get this right. Most of the time, when someone keeps bringing up old stuff to you, it’s because they think you “don’t get it.” Being inpatient with the topic or avoiding it all together only further propagates the problem.

Here’s what you need to do:

You likely have an idea of some specific things you have done that your loved one is upset about. This is good because you can address them directly.
For example, if your wife is really mad at you because you always use to “lie right to her face” about your drinking, then you need to directly acknowledge it. I realize that you both already know that you lied, but you need to own up to the behavior. In addition to acknowledging what you did, it’s really helpful if you can express empathy for how that made the other person feel. And if you want mega bonus points, admit to something the other person didn’t even know (or at least didn’t bring up).

It should sound something like this:

            I know I lied right to your face a bunch of times about my drinking. Sometimes I even turned it around on you by saying something like “You never trust me, I might as well keep drinking.” I remember that one time when ________________________. That must have made you feel like you were going crazy and furious all at the same time. I knew all along that I couldn’t fool you, so I would always just try and deflect the situation. It was pretty crappy of me to make you feel like you were the bad one when I was the one who had done something wrong.
This way of communicating, really helps the other person know that you truly do understand. Which is important, if you want them to stop bringing it up all the time. If you want something to be in the past, then deal with it once and for all and move forward. Avoidance is a sure-fire way to make sure it never goes away!
Make sure you give specific examples of things you did (or didn’t do), and express empathy for the other person.
Addicts/Alcoholics have a magic super power of being able to turn their mistakes around and make the other person feel like they are the one doing wrong. This makes your family feel crazy, confused, angry, etc…
If you want to regain your trust and move forward then you MUST be DIRECT, HONEST and SPECIFIC. Do not avoid these conversations. In fact, if you really want to get the job done right, then be the first to bring it up.

If You’re Super Nervous About It:

If you are worried that the conversation might be difficult or emotional, then you might want to write it in a letter. This gives you a chance to really think about what you wanna say and how you wanna say it. It also gives the other person a chance to decide how they want to respond.