The blower runs constantly for several minutes, with the sparker attempting to light every few seconds through that time period. It's like the flame just flickers in there before going out every time. I feel a burst of heat and that hollow "whooshing" sound that happens when the flame ignites, only for it to immediately go out. Only after several minutes does my unit go into lockout mode.
Yup, this is all typical fail-safe stuff. When you turn on the furnace, the first thing that happens is the blower motor is turned on. Inside the furnace there is a "sail switch" that blows in the wind made by the fan. If that switch does not activate, it tells the furnace controller that the fan isn't running, and it's not safe to fire up the furnace with no airflow (it would burn up). That is check number one.
Then it opens the gas valve and fires the ignitor. If it doesn't detect a flame, it will shut things down. It will try to re-start some amount of tries before it gives up (lock out).
As others have said, either you have a bad control board, or a poor electrical connection to the flame sensor, or there is debris in the furnace causing poor gas flow and making the flame extinguish.
When you open the cover to the furnace on the outside of the RV you should see the burner inlet pipe with the ignitor. Sometimes there are two prongs - one for the spark and one is the flame sensor. Other furnaces combine both into one prong. Make sure the prongs are rust and debris free.
If you remove the bracket holding the ignitor assembly you can usually stick a vacuum pipe up the inlet pipe and suck out any debris down there. Make sure the heater is cold first of course.