Good afternoon Pomology.
Had the same problem with my Atwood furnace a few years ago. Drove me crazy and spent a pile of cast replacing parts that were probably good.
I bit the bullet and went to a friend of mine down on Cape Cod who does R/V repairs of all sorts.
It turn out to be the "High Tension" wire that supply's the electricity from the main electronic board to the spark/sense rod.
What made it worse for me as my furnace always worked fine in the moderate temps and fired up every time. When it got down to say in lower 3o ties or colder - sometime it would fire up - most times it would not.
Hope this helps
BTW - Everytime you shut off the on/off switch on the furnace - you are resetting any faults what may be on the board.
Mike T