Howdy,
#62 (below) is the ESA. It's a collection of electronic components potted in epoxy which will "stumble" the engine when you come out of gear.
#37 is the "Shift Actuator", it moves when you place either push or pull "pressure" on the lower shift cable (using your throttle/shifter).
When in the water in gear, there's enough "pressure" to cause the shift actuator to move and operate the microswitches associated with it when you try to shift from FWD or REV to Neutral.
If it also moves enough to actuate the ESA when going into gear it's possible that your lower shift cable is defective (and is binding). If it's is moving when it should and the ESA is NOT working, the module, switches, or wiring could be defective. If the Shift actuator is frozen, or and not moving that could cause it too. That's why they suggested moving it (by hand)