I tried removing each plug wire one at a time. Each time the engine began to run rough but continued dot run. The Engine Numbers you asked for are:
Model 405877
Type 0010G1
Code 150202YG
hope this helps.
Excellent, you have now ruled out ignition problems so are half way there.
Charge your battery overnight, or hook up a second battery and run the blades.
If the problem goes away then either the fuel shut off solenoid is bad or the charging system is bad and the solenoid is not getting enough juice to stay open.
You can also read the battery voltage as the blades engage, it must stay above 12.5 Volts but it might drop for a brief moment the instant the blades engage the return to around 13 V, that is OK.
NExt test is to get a trigger pack, not a spray can, of WD 40 or similar and saturate the area between the carb & the engine with the blades running.
So you will need an assistant or a bag of sand on the seat.
Turn the blades on & off while you are doing this or manually push the throttle rod on the carb to make it open the butterfly.
If there is a leak then the engine will faulter and should blow some white smoke as the WD 40 gets sucked into the engine.
Try to get to both sides of the manifold.
The plastic ones are known for warping & spliting allowing air to be sucked in when the manifold vacuum increases as the throttle butterfly opens which happens whenever you put a load on the engine, like turning on the blades.
when you turn the engine off & it has stopped running , pull the dip stick tube and look for smoke coming out.
This indicates a leaking head gasket.