With these furnaces, the fan will come on before the igniter lights the gas. When the furnace runs and heats the TT up and get to the desired temp and shuts down, the fan will continue to run for a minute to cool down the furnace area.
One winter camping trip we had the same issue and found out that it was a shorted out battery that could not keep a charge. The converter ran everything (lights, fan) well (with a bad batt) until more steady power was required to keep the gas electronics open. Once the furnace fan was going and then time for the gas to be lit... came on for a short time and then the flame would go out. Had an electrition and furnace guy in the group I was with.... they looked high and low over that thing and everything was 'good' as long as I used an outside batt source. Hooked cables from the truck batt to the TT batt and it worked fine.
When you get a chance have your batt Load tested.
2013 Forest River Salem Hemisphere SBT312QBUD
2001 CrewCab F-250 7.3 PowerStroke Diesel
SuperChip, BTS transmission, 6.0 Trans Cooler