The basics..........
At the powerhead, disconnect the wires that lead to the lower unit, check the ohm reading on the wires that lead ro the lower unit. That should be approximately 8 ohms. The reading on both wires should be identical. Is it? If that reading is as it should be, the electromagnetic coils are okay.
The wires that you disconnected at the powerhead... the ones that are part of the powerhead wiring harness... with the key in the ON position, you should have a full 12v at each wire depending on which gear you have the control set to. Do you? If so, go to the next step..........
Connect those powerhead harness wires to the wires leading to the lower unit and repeat the voltage test as above. Even with the wires connected, depending on which gear you have the control set to... you should still have a full 12v reading at the connection. If a voltage drop exists on this test, the shift switch is faulty or a bad connection exists somewhere between that connection point and its supply point.
If all of this is as it should be..... testing as per racerone's instructions above (#12 reply) would be a good test for the shift springs.
********************
Just for general info on the shift system for you.........
There is a hub splined to the propshaft for each gear.... a heavy duty shift spring attached to each gear.... an electromagnetic coil positioned at the forward and rear portion of the gearcase.
When 12v is applied to the forward coil, it attracts the spring attached to the forward gear and that spring wraps around the forward hub which is splined to the propshaft like a Chinese finger trap which in causes the propshaft to turn. The same scenario holds true for reverse.