1. Is the gas fully turned on?
2. Do you have plenty of gas in the tank?
3. Does the auger turn when you engage it?
4. Does it stop after a period of time if you do not engage the auger?
5. Can you turn the auger by hand when the unit is off and the auger is NOT engaged?
Other than adding a small load on the motor, engaging the auger should have very little effect.
If the auger is binding, it's possible it is causing the problem, but not likely.
One of the first two items is what I would first consider being the problem.
My old Ariens does the same thing until the engine gets warmed up. If I leave the engine running for 5 minutes or so before engaging the blower then it works ok. Using the fuel enrichment button (the carb is from a lawnmower) to give it a few extra spritzes of fuel will keep it running.
Have you tried to keep it running by blocking off some of the air?
Older Ariens use a lever to engage the auger that locks into place. With the auger lever locked to engaged one had to either hold the drive lever down or hold a small secondary lever with an electrical interlock switch. If neither lever was held down with the auger engaged the engine would kill itself.