Hey folks, final update. I ordered a new igniter, new solenoids for the gas valve, and a new control board (dinosaur brand). I decided to just replace all 3 parts rather than keep putting it together and taking it apart again 3 times.
Unfortunately, I broke the igniter getting it in place. The parts people only sent 1 solenoid instead of 2 and they were out of stock. So I polished up the old igniter, replaced only one out of 2 solenoids, and went on to the control board.
When I took out the old control board (which was very easy), I saw that one of the components was melted, with a bare wire sticking out the side of the melted plastic. This seems like a very Obvious sign of trouble, so I'm really glad I ordered the new one.
Once I got everything in place, I put it all back together and turned it on. It worked perfectly on the first try and has continued to work perfectly all week. Perfect timing because we just got the 1st snowfall of the season in North Idaho and my trailer is cozy and warm all night thanks to the furnace.
Thanks for your advice and help!