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Tried to wake up the '68 after its winter slumber.
Got in and cranked it up, no problem with the battery. After cranking for about 15 seconds I turned the key off to give the starter a breather, but it kept turning.
I pulled the negative battery cable off, put it back on and it was quiet. Tried to start again and the same thing happened.
It sure sounds like it. The starter solenoid maybe sticking and not de-energizing. Good thing you can get to the battery quickly to disconnect the power. I installed a on/off safety switch for these type of emergency power issues.
Either the starter solenoid is not releasing when de-energized OR the Bendix drive on the starter shaft is not releasing. Either way, you have to drop the starter & solenoid to affect a repair. And, I suggest that you replace both while you are at it. You also will want to check and/or replace the brushes in the starter motor. There is no need to replace the entire starter motor unless you just don't want the work involved.