It could be the PTO it self needs to have the air gap adjusted also. I am not sure on the 425 but on my 318 the gap should be .018. You might call your local JD dealer and ask them if yours would have the same gap.
425 PTO is driven from the transmission, no air gap necessary. There are many safety interlocks that should be checked first. Parking brake, seat, reverse switch, and then the PTO switch.
I had this very same trouble. Cause was a faulty circuit board associated with ignition switch. There is a relay on this board that activates the PTO solenoid.
If I were you I would by pass the reverse switch and seat switch and that would eliminate both of those, but if you can back up with out the pto kicking out and not pulling up on the switch, you may not have a reverse shut off switch also known as the Rio switch.