Be careful... This is a major way that a fire starts in the John Deere. My X500 JD burnt this way. A fire extinguisher did finally put it out. My neighbor was not so lucky. His tractor burnt to the ground.
I had the same problem as the OP. After cleaning the carburetor several times with no improvement. we found the problem. The fuel filter "in the fuel tank" on the end of the fuel line was pluged. Replacing that filter has solved all the starting/running issues.
We used a stiff wire with the...