Thanks everyone! I ended up getting a total of 5 copies. I figure I'll make some regular photocopies of those when they arrive, and carry one of each when I go to get things updated, and offer to leave the photocopies after showing the certified version if for some reason what I need can't be resolved in a single visit.
I'll let everyone know how it goes once they arrive! Apparently it's marriage license/name change season at the local courthouse, so the usual 2 week wait is currently 4-6 weeks.