Thanks for the diagnostic tips, have not had a chance to try those out yet.
Spannerwrench: I can't tell if the compressor keeps running after the fan stops, because the fan is basically always running. When I say the fan "stops", it's for about half a second - the fan slows a little, but then it clicks back on right away.
I did use an infrared thermometer today to verify. The fridge has been set to 37F in fridge, 4F in freezer for sevreal days. Measured temps were 36-37 in fridge, but way too cold in freezer: -6F to - 12F. The freezer is pretty full, but not so full that it is restricting airflow, I think. I will beg my wife not to restock it quite so aggressively and see if the freezer side stays that cold.