There is an old addage that goes like this
90% of ignition problems carburettor and 90% of carburettor problems are in the ignition.
When you turn off the ignition, the fuel solenoid shoves a plug into the main jet pick up tube.
So for an instant the engine is running lean.
This is the instant of good running you hear just before it stops.
It also tells me that the magneto is not shutting off and the engine is starving off.
So you have the jackpot , both a carb & an ignition problem.
Carb is flooding so it needs a clean and perhaps a new float needle & seat
Photo instructions for that are here
http://outdoorpowerinfo.com/
Also your kill wire is not grounding the magneto so pull the blower housing off and check the thin coil wire.
It should be open circuit to ground with the ignition on and closed with the ignition off.
Being this is the beginning of the season chances are some thing has had a munch on it during winter.
Shame on you for not feeding your local critters over winter :laughing: