Eight hours later, I think I've figured it out. I'm thinking that every time the group syncs back to Constant Contact, they're processed as being re-subscribed and therefore getting the welcome message again. Can't believe I wasted a day figuring this out, but better later than never.

Note to future module users: Do not sync the group that your subscribers are added to. Review it regularly, and assign them to other groups for email distribution.

Note to developers: if there is a way to sync the group and avoid the confirmations, that'd be great too