Disconnect the coil kill wire on each coil and see if the engine now runs on both cylinders. The engine can be shut down by the after-fire solenoid on the carburetor, when the key is turned off.
If it does run fine disconnected, the kill wire harness has a bad inline diode for one of the coils.
Use the numbers stamped into the top of one of the valve covers to look for parts. Parts Tree, Ereplacement.com or Jack's small engine should show the kill wire harness but be careful of which one you have, because they had a couple of different ones.