It seems unlikely that you'll meet anybody, and even less likely that they'll inquire about what you're buying. If they do, then just be ready with a simple and plausible cover story--like, "Oh, I'm headed to a friend's house and they asked me to grab a few things for them." Don't over-think it.
And, since you mentioned that you don't live alone, I'll add that I think buying at the store and having total control over when they show up at your place of residence is probably the best form of discreteness you can hope for. IMO.
I was buying Pampers with my allowance when I was 12 or 13 years old. It's a fear worth putting behind you.