Gosh, I think you need to pray about this. Had a similar thing happened to me I would let my children know. It would not bother them to any great extent and they would be pleased on two fronts, my honesty with them, and the fact that I had done something about it and was back in recovery. They may be hurt if I kept this from them and they found out some other way.
Secrets in families are not a good thing, and they have a way of getting out. I am completely open with my family.
If I happened to feel as you seem to be indicating I might also take a good look at my real motives. Shame seems to be a thing. I could be ashamed of such a situation especially as I might feel I had let people down. Would it be a good thing to internalise that shame? Could my ego be getting in the way here? Is my real reason for keeping the secret that I feel my children will think less of me if they know the truth? Would I be playing God here, prejudging how they will feel and react?
The situation you described seems to be common enough among alcoholics, it comes up in meetings now and then. It seems to me the ones that are honest and open about it not only recover, but set a great example to anyone else of the rigorous honesty this program requires.
I am not suggesting any other course of action other than you pray about it and consult with others close to you. All the best.