Yesterday, we cut away the offending drywall, but did not find the source of the moisture. The plastic moisture barrier over the insulation was still intact (except where I accidentally cut it) and showed no signs of moisture on the insulation side (which I could check due to my accidental cuts. Similarly, all of the studs and flooring did not show signs of water damage; although, there was some mold at the base where the damp drywall met the subfloor. There are no heating pipes in that area either (which explains why the closet can get chilly in the winter); however, now I'm not sure how the pipes get to the boys' room.
In short, the damp and mildew is still a mystery. We measured out the are that I had removed and went to Home Depot to pick up supplies to fix it (we also checked out some other project materials that we could use), but we left that project for today. While working on it today, I noted that one of the wall-to-wall shelves in the closet was not cut correctly and thus did not rest properly on its supports. As it was also narrower than we would like, I made another Home Depot run to get another board for it and the one above it (which was cut fine but was even narrower). I then realized that the offending boards would be about the right size for the pantry shelves in our kitchen (which does not have enough shelves).
At Home Depot, I had them professionally (and quickly) cut the new boards. Only on trying to fit them in did I realize that my measurements (which I did do twice) were apparently off by a quarter inch. After a slightly botched job using my jigsaw on one shelf (it still fits much better than the board it was replacing), Janelle had me use her circular saw on the second one. It turns out that I can screw up with that too, but not as badly. Speaking of screws, I did take the precaution of drilling and then screwing the new shelves in place--something the old shelves did not have. As long as I had the circular saw out, I used it to cut one of the older boards into the correct (if not perfect) size for the kitchen pantry.
Putting in the closet shelves might have scratched up the walls a little bit, but I also did not plaster or paint the new drywall I installed (as we want to see if the damp returns first). It's in the far back corner of our most inaccessible closet, so I'm not overly concerned about how it looks at the moment. Still, we will probably need to paint it in the future. Meanwhile, I still have two shelves to put into the kitchen pantry. However, as this will require more measurements and drilling, I'm going to leave it for another day.