My brother-in-law, who is good with things like this, offered to take a look at it for me. So I hooked the trailer up and hauled it down to his shop. He looked at it for a bit, popped the top off the stove for a minute, put it back together, tried to light it again, and it lit just fine. DOH! Now he knows I'm an idiot. LOL
The stove, oven, and furnace all work fine now. My only guess is that moving it jiggled something so the propane started flowing again. It makes me nervous that it will happen again, but I know the remedy now - just drive it over a couple of bumps.
I'm off in the morning to visit the Black Dragon Wash in southern Utah (just off I-70 a little ways west of Green River, Utah).
Thanks again for everyone's help!