h20ski, I was not hooked to 110. The fridge has on/off and auto/gas switches, so I set the fridge for gas and then pressed on. It took 30 to 45 seconds of trying before it was able to stay lit, and then I could hear a constant, burning-type sound from inside and outside the trailer. It could still have been the 12V driving it, except it sounded like gas was burning. I assume (and I'll try it in a little while) that the refrigerator hums when it's running on 12V/110V power.
Yes, I've tried turning the valves slowly multiple times (with over 5 minute breaks in between). I watched the guy fill the tanks up - they were about 54 lbs. each when he was done. I even tried opening the bleeder valve briefly on one of the tanks, because I was told that can unstick the valve. I haven't tried to set the tank selector before opening the valves, and I'll give that one a try.
The furnace blows cold air for three or four minutes, and then it gives up. The furnace sounded like it was trying to light gas about three times early in the cycle. There isn't a pilot light or anything I need to set on it (besides the thermostat) is there? I thought they were all auto-igniting for safety these days.
The propane detector (next to the floor in the kitchen area) is showing a green light. Could it be faulty? Would it shut off propane to the stove and oven but not the fridge? And I haven't checked the fuses, because I didn't think that would affect the stove, but could one of the fuses affect the stove?
It's starting to sound like a blockage to me, but it's weird that the fridge runs and not the stove, oven, or furnace - the fridge is furthest away (physically) from the propane tanks. I haven't tried the water heater, because I don't have water in the trailer yet. I'll take a peek under the trailer to see how the propane line is routed, and maybe I'll find a clue there.
Thank you everyone for your quick responses!