I assume it is a Direct Spark Ignition type (As opposed to a pilot light type) What you describe is a flame sensor issue.
There is a rod, that is heated by the flame, Depending on the control system it may also be one of the ignition "points" inside the burner assembly.
If the burner is actually lighting (you feel hot exhaust) then shutting off it means the control board is not detecting the heat (Flame) via the thermocouple.
This could be due to dirt build up on the Thermocouple insulating it to the point where it does not get hot enough fast enough (Or so I'm told)
It may be due to a poor electrical connection, either ground or between the sensor and the control board.
On my furnace it was the board.
IF... It turns out to be the board... Contact Dinosaur Boards, IF, and that's a big IF, they make a boad for your unit (And you can identify it well enough for them to figure out which one it is) They make very good, very well buit boards. Lower cost too, kind of poofs the "Made in china is cheeper" folks since thy make here in USA far as I know.
When I say cheaper. Think 100/250ths. (That's Their price/Attwood's on the board I needed when I needed it. Prices subject to change or differences depending on the board you need.
Home is where I park it!