OK, you need a basic understanding of how it works. When you push the reverse button, battery voltage is sent to both the green and blue wires going down to the lower unit. That is why you need to check that before going any further. It would be futile and foolish to tear into the lower unit when the trouble is in the control.
Moving on from there, the voltage at the motor energizes both solenoids in the lower unit. Those solenoids close two valves in the oil pump in the nose of the gearcase. That is as far as the electric part goes. The actual shifting is done by oil pressure from that pump moving a piston that moves the clutch dog between the forward and reverse gears.
So............If we establish that the voltage is reaching the solenoids, then either the reverse solenoid is bad (probably not) or the control valve is not seating (probably not that either), or the piston is excessively leaking or is partially stuck in it's bore. We know the neutral solenoid and it's associated valve are ok because you have neutral.
Bottom line is go at it in order. Check voltages, check solenoids, then into the gearcase if necessary. If you have to get that far, it is a major job requiring total teardown. Some tools are required. Factory shop manual highly recommended.
