Keep in mind it's been close to 5 years since I did much service work and my memory is getting a bit foggy.

Which fan is it you are talking about?  If I remember correctly that furnace probably has a combustion air fan ( a small fan blowing air in to the combustion chamber) and a circulation air fan (blowing air through the house).

The combustion air fan should act just as you described.  It should kick on when the thermostat calls for heat, purge the combustion chamber of any gas setting around, then when the ignition starts up it would continue until either the burner lights OR if it does not light would purge any fuel gas from the combustion chamber.

The circulaion fan usualy operates ttally independent of the other.  It turns on when the temperature in the plenum is high enough to need heat moved and turn off when it cools down.  If the air returning to the furnace is particularly cold it may cycle on and off repeatedl while the burner is going.

ALSO I would suggest you look at this recall notice, also listed here,  then contact the number given.