Since she looks and dresses like you, you might want to tell her how great she looks...
Funny how easy this was growing up, now we put way too much thought into it. Just tell her you love her hair or her outfit and where did she get it (done)? Always a great conversation starter with women. Tell her you bought too much groceries and maybe she wants to join you for dinner.
I don't think you have to be wary of the conversation then turning to your shared sobriety. Maybe when dinner's ready, you mention not to have wine with it because you quit, ask what she likes instead. Most newly sober people start out feeling lonely and having to learn how to make new sober friends, chances are she's no different!
And, people gossip, they just do. If I were her, I'd rather have someone ask about personal things straight up out of curiosity, than make assumptions behind my back. Do you see a lot of friends coming to visit her, or does she go out often? Sounds like she's going through a lot, she might really appreciate your effort, and if not, no harm in trying, right?