I had a similar problem two years ago with my PTO solenoid. At the same time I had a 30 amp voltage regulator, original equipment, that kept going bad. My PTO solenoid problem did not go away after I solved the voltage regulator problem. I assume the ignition switch would heat up and dropped the voltage to the PTO solenoid. Rather than replace the switch, I installed a wire with a fuse from the battery post on the starter to the PTO switch. No more PTO solenoid problems. Yes , the PTO solenoid switch will remain energized with the ignition key off. I kept the wire from the PTO switch to the ignition switch that prevents the engine from starting with the PTO switch ON.
Hope this helps,
Dale