I installed replacement motor and grommets. After figuring out this was the most likely cause of the problem, I watched a couple of videos. I didn't find any videos featuring a fridge like my 2003 model, but I was able to figure out my way back through the freezer compartment with trial and error until I exposed the old evaporator fan and motor. Tip: pliers can be very helpful for releasing the clips of electrical connectors - I had to unclip several.
As soon as I got to the fan, it confirmed that this was the problem: the "propeller shaft" had a huge amount of play as it rotated, causing the fan to rub against the housing as it turned. Sure enough, as soon as I had reassembled the fridge with the new parts, it worked perfectly with no noise.