It sounds like there is an air leak in a hose somewhere in-between the tank and the injection pump. I'd check your hoses and tighten the clams, then bleed the system.
Thank you for your idea on the fuel shut off solenoid you don't know if that is what John Deere calls a fuel shut off timer and how to check it,I think that we are on the same page
it almost sounds like the key switch isn't continuing to send signal to the fuel solenoid
to keep it running after release from the crank position,
maybe that can be tested....