OK, I think I got a handle on what might be going on. I took apart the cooler again, took out the cigar boxes, and noticed a little standing water in the back of the cooler. This cooler doesn't have a drain. What's happening, I believe, is that the air blowing past the heat sink is cooling the air and condensing a lot of the moisture out of the air, and the air being blown forward is drier than the air being sucked in. For whatever reason, the air in the back of the humidor is not really reaching the front, so there's a localized pocket of high humidity.
I have thought about buying some kitty litter and putting it in small bags to absorb the humidity in the back. But there still could be issues with mold, I suppose. This is turning out to be a lot more work than I planned, and given the relative low cost of my cigars, perhaps not worth it.