I would never volunteer that I'm a recovering alcoholic. Sadly, there is still too much judgment, and most of it is negative.
BTW I wouldn't volunteer info about medical conditions, quirky habits, outrageous relatives, credit card debt or stupid stuff I did as a teenager, either. IMO opinion a professional environment demands a degree of reserve that seems to come naturally to you. (You don't have to worry about oversharing like some people do
)
One thought is for you to actively seek out new interests...and then you'll legitimately have something interesting o talk about in casual conversation. Your new hobby/interest does not have to something mega social - you might like learning to play guitar, or raising zinneas, or petting the animals at the shelter rather than joining a bowling team. Pick something you enjoy!
Sorry for such a long response. You sound like such a nice person, and I'm giving one more vote for keeping your challenges with alcohol private from your new employers. (at least for now, and if they don't directly ask)