Let's put this one to bed:
I got a gas valve kit, replaced both valves (took all of 2 minutes, once I found my short Phillips head screwdriver), and it's done 6 loads of laundry without a hitch. Given the ohm readings on the 3-prong coil, I expect it was the busted part. The 2 prong part seems to ohm out OK, but I didn't leave it in since every site I've read says change them out as a pair.
I'm still going to watch the dryer for a while because I don't believe the ohm readings from the thermostat. But that part is just as easy to replace if it should prove bad...
Thanks denman! 
Jonathan | 
