Hi and Welcome!!
This section is fine!!
For your problem, it would seem that the igniter/flame detector in the furnace is not functioning as it should. (This part does the both jobs.) You may have to remove the furnace to do much but I would check the wiring between the detector itself and the wire connection. Trace the wire and remove the connection at the circuit board and clean it well, both the wire end and the connection on the board, and then reconnect. You may have to replace the igniter/detector if cleaning and correct placement fails.
Theory: To detect the flame the igniter/detector has to sit in the flame. In the ignite mode, the igniter will flash 3 times and ignites the gas. When the flame ignites, the detector portion generates a very small voltage that is picked up by the circuit board during that 5 second period. If the flame is not detected then the furnace (or any other gas fired appliance) will shut down as a safety precaution. Since the voltage is very small, any corrosion etc. can prevent this happening.
You should find this article from the library helpful...
http://www.rvforum.net/joomla/index.php/16-rv-how-tos/333-furnace-troubleshooting
The fridge, the water heater, and the furnace all function similarly when operated on gas (propane).