The solenoids are probably the least thing to be a problem and the last thing to check.
First thing to check is the oil in the lower unit. The oil is also the hydraulic fluid for shifting, so if it is thick, dirty, full of water, whatever, you have problems.
Second thing to check is voltage at the blue/green wires at the powerhead where they go down to the lower unit.
Check voltage there on each wire, with them still connected. Should have battery voltage (12v) on green in neutral and on both green and blue in reverse. The push-button switches are somewhat troublesome.
I have already ran a continuity check and have good volts and grounds running from the switches to what appears to go down into the lower unit heading for a solenoid, and have replaced the lubrication with type c fluid and found no metal flakes and no sight of water.
If you haven't already found this out, it has to be running to shift.
