OK, we need to determine if it is an electrical or fuel problem.
So remove the blower housing and remove the kill wires from the coils.
Start the engine then shut it off a few times.
If the engine shuts down every time, then you know that the fuel solenoid is working.
Replace the blower housing then go mow.
Be extra careful as all of the safety circuits run off the kill wires.
If the problem goes away then you have a problem with the mowers wiring, let us know & we will walk you through finding the fault.
If the problem continues replace the kill wires then remove the fuel shut off solenoid on the carb & replace it with a short bolt.
Check that the mower shuts down when you turn it off then go mow.
Problem goes away you have a fault with the solenoid or the wiring to it, this is very common.
If the problem persists then find a replacement fuel tank.
You should be able to get one from a mower repair shop ( not a glass front retail store ).
Hook it up and go mow.
Try it both direct to the carb and also to the fuel pump
Problem goes away then you have a fuel supply problem.
Could be bad lines or debris blocking the fuel tank outlet.
Problem persists then you have an internal engine problem or blockage in the exhaust.