I have a Chinese knock off of a Woodward on my Branson. The replacement parts are real Woodwards, and the prices are breathtaking.
These are sealed dual coil (winding) solenoids. One winding is strictly for pulling the solenoid in. The second is a holding winding. A timer on the tractor drops the current to the pull-in winding about one second or less after it is first applied, when the ignition is switched on. The holding winding then keeps it in until the ignition switch is turned off.
If the pull-in winding is energized too long, it will burn out. If the burn out is caused by an overly long timer period, or one that sticks, the replacement solenoid is likely to burn out as well. The pull-in winding draws a very heavy current, as high as 41 amps on some models.
In the (good) old days, diesels didn't have this sort of expensive nonsense. The fuel shutoff valve was held open by a spring or cable friction. An attached cable with a knob or handle ran to the operator's position. Shutdown was done by pulling out on the knob or handle until the engine died. Simple, effective, and the engine required absolutely no source of electric power to keep running once started. If there was no spring, you had to push the knob back in to be able to start the engine.
Woodward has a solenoid catalog available
here. The 1503 is no longer listed. It may be available to OEMs on special order.
Woodward solenoids are sometimes listed on EBay.