You have the choice to embrace sobriety. To live sober and secure in yourself.
While I would never not nod my head and say hello in passing to someone, there are absolutely children mine are not to play with at their home. We find very polite ways to decline invitations and fortunately we are a busy family.
However, little ears sometimes hear things they shouldn't. It may not be that the mother told her child you were an alcoholic, it may be little ears overheard and the slight might not have been intentional in town. My husband and I both grew up with parents who were employed in jobs requiring community involvement, my ears heard a lot of things they shouldn't have. Add in being related to half the town... slim pickings when I started dating. I never put out on a first date - might be my cousin, best to get all clear after the date, before a relationship LOL.
Don't let it bother you, don't let it affect your sobriety. As parents we should never put adult matters on the shoulders of children, but sometimes it happens unintentionally.
We got lucky with my step son, he had no clue about my alcoholism. Until recently, when he asked if I don't drink and I said no and we had a long overdue talk. His comment, I knew you were a workaholic, didn't know an alcoholic too. I was like, why did think there was a couch in my office? My little ones are blissfully unaware, except Mommy is happier now and not so tired.
Dh and I are very careful to have conversations in our garage, alone. That is where we go to fight too. When it isn't a simple disagreement. We took that from our parents. We heard too much as children about matters that should not concern us and little ears have bigger mouths
Good luck to you, you have a chance to change everything.