I first checked and cleaned the vent pipe leading outside the house. It was actually pretty clean for being in use for almost 20 years.
I then disassembled the dryer, referring to a couple videos on the internet.
I tested all the components and all tested OK. I noted that sometimes the Gas Valve Coils will test OK, but still not work correctly. The propane burner would light, run for about 35 seconds, then shutoff. It would try to re-light but never would.
I spoke with tech support and eventually ended up ordering a Gas Valve Coil Kit and a Gas Valve (in case the valve itself was faulty). It turns out that I only needed to replace the coils, not the whole valve.
While I had the dryer disassembled, I cleaned up everything and blew compressed air through the motor to ensure it was clean internally. I then reassembled the dryer and tested the propane burner before doing final reassembly. I noted the burner and related sensors seemed to run better once the front cover of the dryer was back in place (there is no inspection port on the dryer).
I've run about 5 loads of laundry through the dryer and everything is working as designed.