Mike writes:

"but the starter does not turn and the system voltage drops only about 1 volt (from about 12.7)."

Based on your reported observations your starter does have a problem, it is not getting the current it needs. I don't know where you are measuring your system voltage but IF is is across the battery then one of three things is happening.

One. The starter is shorted and when you close the solenoid 300 to 500 amps circulates through the battery dropping the terminal voltage to 1 volt. (very not likely).

Two. The battery has an internal, intermittent and temperature dependent open in one of the cells or intercell connections.  (not likely)

Three. One of your connections is corroded and when things get hot it expands and opens. (My first bet) Remember, when you heat a fastener everything gets bigger, even the gaps.

If I were you (poor you) I would check the ground connections between the engine and  battery and then all the other high current joints. Are your high current ring terminals crimped and soldered or just crimped?

Regards
Brent Regan