Forgive AND Forget?

I've thought a lot about the phrase "forgive and forget" lately. I personally find it difficult to do. Forgiving is something I do not struggle with - it may take some time and some work, but eventually I reach the point of total forgiveness, and I hope I deserve the same in return. Offending anyone, for any reason, is definitely not part of my life's agenda. I don't mean to do it. And, when needed, I always apologize for the wrongs I've committed. Actually, many times, I over-apologize which usually makes things worse. Since I was born I've wanted to be a friend to everyone, and make everyone I come in contact with happy. This results in a lot of unnecessary stress that I must deal with. Even when I am not outright responsible for some occurrence or activity, I take it upon myself to make things right, even if I have done no wrong whatsoever. I've struggled with this my whole life. I am very appreciative that most people have been forgiving towards me throughout my life.
Forgetting is something apart from forgiving. I think we try to forget too much. How are we to forget when someone offends, or when something happens? Wouldn't it be nice to simple erase our mental hard drives and honestly forget that something happened? Deleted. Recycle bin. Gone. Our minds don't work this way. There is no limit to what our memories can hold. Images can stay fresh in our minds throughout our entire lives. I think there is a reason for that.
Although it is true that our bodies make us up physically, it would be nonsense to say that is "who we are." We are much more. We are made up of everything we've ever done, said, thought, seen, etc. Experience makes us who we are. If we've done something we shouldn't have done, we remember it so we don't do it again. We shouldn't delete it, or we'd probably fall back into it over and over.
I guess what I'm trying to say is that we shouldn't try to delete our memories or things that we have been a part of. I'm certainly not proud of everything I've done, but I'll always appreciate what and who has made a significant impact on my life. Those are things worth remembering.