[RichardJ]

>> the starter stays on, continuing to turn the motor over after the ignition kicked in.<<

By that, do you mean to say the starter continued to crank when the engine started and you returned the key to the "Run" position.

The starter solenoid has two coils. A pull-in winding and a hold-in winding. Together they draw 42 Amps, but only for a split second when you hear the normal starter Click.

After the Click, the engine is cranking and you are holding the key in "Start" position, only the hold-in winding is energized, drawing only 10 Amps. At the click, the larger pull-in winding should be taken out of the circuit.

Unless something goes wrong in the solenoid. Then the starter circuit draws a continuous 42 Amps and the whole starter circuit gets very hot and lets out the smoke.

When the copper is heated like that, it oxidizes every wire crimp and copper connector surface in the entire starter circuit.

When two wires plug into each other, you have two wire crimp connections, plus a friction connection where the two copper connectors plug into each other.
I didn't label all the possible corroded connections, because you could have 20 points with increased resistance in the simplest starter circuit.
A small resistance increase in several of those connections can combine to a large increase in resistance in the entire circuit when under load.

Checking the voltages when NOT under load will be very deceiving. In an open circuit, there is no voltage drop across any corrosion resistance you are looking for.
An ohmmeter will be of little use as well.