Estate Gas Dryer by Whirlpool, Model TGDS740PQ1

Dryer runs fine 95% of the time. Dried 6 loads, one right after the other,
with no problems today.

The problem is that on rare occasions, it will stop before the drying cycle
is over and I can't restart it right away. If I wait until it cools down
and press the Start button it will start up and complete the cycle with no
problems. However, if I press the Start button right after it shuts itself
down, here's what happens:

With the timer control set to any of the heat cycles, (high heat auto, low
heat auto or timed dry) I can hear a relay or solenoid actuate and i can
hear a hum from the unit while I hold the switch in, but the motor doesn't
start turning.

With the control set to the Air Fluff Only setting, there is no relay sound
or humming, just a whole bunch of silence.

I did one more test: when the dryer was running properly and in the Cool
Down period of High Heat Auto, I turned the timer control back into the
area where the heat would be on. When I did this I heard the same relay
sound that I hear when Start button is pressed but the unit doesn't start.
A few seconds later I could hear the gas turn on.

I believe that there is a pair of solenoids for the gas valve. Is it
possible that the noise I'm hearing is one solenoid activating but the
other one hanging up and humming? But if that were the case, why wouldn't
the motor start, especially in Air Fluff Only mode?

Darn, thought I figured it out while I was typing this, but now I'm not so
sure.

Any idea what to look for? Thanks!