Couldn't decide whether it was the control module or the thing on the bottom/back of the frig where the water comes in. Based on the AppliancePartsPros website, I went with the control module. It arrived in less than 24 hours (paid for standard shipping!) and followed the video on the website to install. After an hour or two, ice cubes galore!!
(The time to do this that I put below was a little extended as I have a bad shoulder and was having to reach up/in to get at the thing.)