Started the car at work yesterday and the starter stayed engaged and kept cranking.  I’m hoping I got the engine shut down before I did major damage to the flexplate.  The starter finally stopped cranking after I slammed the door shut.  I was able to fire it right back up with no problems aside from an unrelated alternator belt squeal.  Should I just change out the starter or should I look into both a starter and ignition cylinder?  Car is a 1968.

Could be the starter stay engages if shimmed too tight and cannot release , the solenoid inner contact terminals had weld themselves together or the ignition lock cylinder could have binded and kept the switch in the START position.

You can interrupt the ignition lock cylinder start circuit by shifting out of park or neutral (provide your have not bypassed it) and that might isolate a cause or rule one or the other out.

I'm thinking the ignition switch, but to keep cranking after shut down is strange.
I brought my '62 Impala home from storage last week and it did the same thing when I started it after sitting for 8 months. The big difference is, it would stop once I moved the switch to off.
Either way, it sounds like condensation/corroded contacts in the ignition switch. I'm thinking I will need to replace mine too, if it continues to be an issue.

Been too long since I did this work for a living in the seventies to recall if a stuck solenoid can continue to cross power the starter from the positive post internally, after the power source from the ignition switch has been turned off, to create this situation.

Larry Lucast wrote:

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My vote is for the starter and/or solenoid sticking. When you release the key, the crank circuit stays energized because the solenoid is stuck in place. If it was the ignition switch, releasing the key physically moves the contact away from the start position.

Or maybe not.

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 I wasn't certain on that, but there were some cobwebs vibrating to that effect. 1978 was a looooooooong time ago for my memory...