Replaced ice maker which was about 6 or 7 years old. That solved the problem immediately.
The hardest part of this job was melting the mountain of ice that accumulated in the freezer. That took about a half hour with a hair dryer. Replacing the ice maker took less than 5 minutes and a 1/4" nut driver was all that was needed. I also ordered a fill valve for $20 but haven't installed it yet. Obviously the fill valve wasn't the problem but I'm going to replace it anyway. Looks like it will only be a 10 minute job too.
Thanks AppliancePartsPros.com!