Nice dryers!
Nothing like a vintage dryer to actually DRY your clothes, I say!! LOL What a nice pair of vintage Frigidaires you have there, Pete. You'll still be using them years from now, I am sure!! And the cats like, which always a bonus.
My 2 cents on the automatic drying - it sounds like something is causing the timer to stop running when the 'automatic' cycle is selected; a thermostat would normally take over until it hit the high limit, then the timer would kick in again for the 'cool down' phase of the cycle. That's what I learned (mostly the hard way) from GE and GE clone auto dryers...
I am really surprised that there is no obvious moisture or temperature sensor - could it be inside the drum??