There's no mold outside the drywall.
I pulled down all of the drywall inside the closet, and there isn't any sign of moisture anywhere. This is a good thing, but confusing. It means that all of the moisture inside the closet, and there is evidence of a substantial amount, came from inside the closet. A closer look at the ceiling area reveals a large, insulation free, gap right in the front corner of the attic. Apparently, moisture has been evaporating from the coats, wafting upwards on drafts of warm air, coming in contact with really cold drywall, and condensing on the surface. This also is a good thing, because it is easy to fix: proper redistribution of the insulation above the closet=problem solved. While I'm at it, I will take the time to insulate the walls as well, which will improve the overall comfort level inside the house.
Now the hard part: actually doing the work.