It is likely air in the line leading from main to heater. I suggest haveing the propane company send someone back out because their service call is much cheaper than a plumber's. Relighting it yourself based on instructions gleaned on the net is not a good idea. .
Time for a story about tinkering with propane gas. Man jumped out of a plane for his first parachute jump. Unfortunatly he wasn't paying attention when given instructions earlier about operating his chute. Just as he was giving up hope,he sees another man flying up toward him from the ground. As they passed he shouts "hey do you know how to open a parachute? "No,comes the reply,do you know how to light the pilot on a propane water heater?"

Thanks for the info that the gas company will be cheaper!

I don't need the pilot to be lit (I don't think?). The flame will come on--it just won't stay on long enough to fully heat the whole tank.....we each get warm-enough showers a couple of hours apart. I guess it runs until a burst of air passes through, causing it to go out? And will only do it once an hour (judging by the code).

Thanks.

As the propane gets low in the tank the gas pressure gets low and the water heater will build up with soot to the point it impedes proper airflow and heat builds up. On certain models it voids the warranty. In some cases the ventilation pipes need to be replaced. Running out of propane is a bad circumstance for propane water heaters. My guess is your problem is soot and venting causing the systems safety mechanism to be triggered. It needs to be cleaned or replaced. roper airflow

Lost prime, not likely clogged. Air needs to be purged out.