So thanks to a suggestion to buy the solenoid outside the dealer, I looked at the solenoid and it is a Woodward Solenoid. This is a model 1502. I found this on their website:
Table 7. Linkage Adjustment
Examine the linkage and look for signs of corrosion, rust, oil and dirt, all of
which can contribute to unacceptable performance of the solenoid. Make
sure linkage moves freely without binding. If the linkage feels tight, check
for bent linkage, misaligned bracket, improper spring on control lever,
swivel head hitting control lever, etc. Loosen linkage as needed.
The linkage and the plunger must all move freely with your fingers.
Turn the Key Switch to “Run/On” position.
Take precautions during these checks to prevent pull coil burnout.
For internally switched solenoids, manually push the linkage all the way in.
The plunger should hold in place. If the plunger does not hold in, measure
resistance across the positive and negative terminals. The reading should
match the hold coil resistance listed in the Solenoid Coil Resistance
Check section. If the reading is not correct, the plunger is not activating
the internal switch and linkage adjustment is too short. Adjust linkage until
the plunger holds in.
For externally switched solenoids, manually push the linkage as far as it
will go. The plunger should hold in place. If the plunger does not hold in,
linkage adjustment is too short (for push type solenoids the linkage
adjustment is too long). Adjust linkage until the plunger holds in.
Manually move linkage from shutdown to maximum fuel, ensuring
solenoid spring will return linkage to shut down position and full travel of
linkage is smooth, uninhibited, and does not exceed specified stroke.
When a connecting rod is used, the stroke is adjusted by turning the rod
on its threads and locking the rod in place with a lockwasher and nut. The
solenoid should be energized during this adjustment. Use this procedure
with ES solenoids only if the linkage length is incorrect.
When bead chain or cable is used, the solenoid should be energized and
the bead chain or cable length adjusted to give the desired lever position.
Plunger travel must be checked, especially when a bead chain or cable is
used in a connecting device. The plunger travel must be limited to the
solenoid’s rated stroke when it is de-energized. An “L” bracket can be
used to limit the plunger travel.