Hi!

The
dryer spins, but never fires up the gas. I never paid attention to it
while it was working, so I don't know whether you can hear a solenoid
or valve open to start the gas, but I can't hear it now that I'm
listening. No gas smell either, so it's not open but unignited
(fortunately).

When the dryer was working, you should have heard a noticeable solenoid
click followed by a roaring noise.

I had what is probably a similar Whirlpool gas dryer get rather wet in a
basement flood. After the flood it would not fire the burner reliably. After
a while it wouldn't fire at all. Taking apart the dryer revealed that the
igniter was working, but that the click of the solenoid was missing.
Eventually something in the dryer must have decided to "give up" and the
ignitor would go out.

So after disconnecting power and shutting off the gas I had a look. The
solenoids on the gas valve (lower left front of the dryer) had removable
coil packs on them. I tested both and found them to be bad. After replacing
both the modules the dryer works again.

It is a guess, but a pretty fair one that your dryer has the same problem.
Remove the coils, take them to an appliance shop and they'll know what to
get. Unless something got way too hot, the thermostat module you found isn't
likely to be bad. I can get the model # off of my dryer if you'd like.

William