After a bit of internet browsing, once again I was led to AppliancePartsPros.com, which help me determine that it was most likely the damper assembly gone bad. Looking at the internals, I saw the issue: plastic gear assembly gone bad. The refrigerator is about 9 years old, and I had heard some knocking, strange sounds for a while now, but thought it was something in the ice maker, and it only happened occasionally. The damper was indeed the culprit, and replacement, while not straightforward, wasn't too bad once I found a video on how to remove the filter housing also, as this is part of the assembly. There are several videos on replacing it if you don't have the filter. Here's the two videos I used, and remember you'll need to turn off the water, and unplug the fridge first. And be very careful when removing the sensor/capillary lines that are in a hole going to the freezer. Also, it doesn't come out of the housing as easily as it does on the video, but with some patients and persistence, it can be pried up.
Damper assembly: https://youtu.be/AS3EAYjIDmY Filter assembly: https://youtu.be/HROBLi9TEgI