OK, I think I figured it out. The drain hole below the narrow aluminum pan (that sits below the coils) and the pan itself were completely filled with ice. Tried chipping some of the ice out of the hole and then dumped a 1/4 teaspoon of salt into the defrost drain. That worked really fast as by the time I'd heated up some water and poured it into the pan, it was already flowing to the pan. I poured a good two cups of hot water through the drain to make sure it was flowing freely and that all the salt was washed out. Put it back together and will keep my fingers crossed.

Update: three days later and the refrigerator shelves have remained dry. i think it's fixed.