I stayed at the shelter (Birch Springs Gap) at #113 20 years ago on my first backpacking trip in the park. It was in March and the trails were exceptionally muddy, the temperatures were hovering around freezing, and the fire ring area immediately in front of the shelter had standing water overflowing from Birch Spring making it seem like the spring was pumping water into the shelter itself.
More recently I did a Fall loop from Twentymile up and over into Cades Cove then back over Gregory Bald. Route was twentymile, long hungry ridge, forge creek, rabbit creek, hannah mountain, gregory bald, wolf mountain. We stayed at 12 and 14 on this trip. We originally planned at staying at 12, 16, 13, but cut our trip a day short when we arrived at 16 and the source was dry. Instead of hiking back up to 15 which had plenty of running water (but was closed due to bears), we pushed on to 14 only to find the water source past the campsite was dry as well. I climbed up the draw for a while until I found a tiny trickle and exercised much patience to fill up there. When we got up to Gregory Bald the next day, the source before you get to 13 was dry as well. We could have hiked over to Moore Springs, but we had already had enough of the unseasonably warm weather and dry conditions so we hoofed it on back to the car.