Update: Ran all the measurements suggested in earlier post and everything is within spec numbers given. Continuity for all connections, starter and compressor, no short in compressor and resistance numbers for started and compressor test okay. There's nothing wrong with the starter and there's nothing wrong with the compressor, according to the numbers. Another thing I noticed is that after pulling the fridge out from the wall, the clanging noise of the compressor has stopped. Could it have been a leveling issue? The fridge is cooling, no strange noises, so we'll call this one okay, for now.
However, the matching upright freezer in this duo (Kenmore Elite # 253.44733102) is now doing what the fridge was doing in that I hear the click of the starter and a hum from the compressor trying to start and then, after a couple more tries, the compressor starts. This is the more typical starter issue I have been reading about. The freezer is cooling but I sense there is a bit of a rise in the temperature. Ice cream is not melted but a bit softer than normal.
Anyway, I started looking for the starter (#
297051000, I believe) for the freezer, just in case, and guess what? No longer made by the manufacturer. Discontinued! Is there a work-around for this type of issue?