I'm betting on corroded connections
I just put brand new solenoids on mine... it works great unless I bury it in muddy water- then the cable connections at the solenoid pack get filthy and it works about 1 or 2 more times. Seems that the relatively high current arcs at the connections, which oxidizes them, and prevent further current flow (but it still goes "click")
clean the solenoid pack, and take a wire brush to the connections. Then reassemble it. I"ll bet it works.
Also, if you want to just test it out you can skip the solenoid packs entirely- make sure that the ground (from winch motor to battery - ) is good first, then you can run a jumper from A to F1, then run + to F2. It should spin the motor.
To spin the motor the other way- you'll want to jump between F2 and A, then run + to F1. I don't recall which one is "in" and which is "out", but you'll be able to tell pretty quickly.
For the test, probably easier to do if you disengage the clutch first so that there is no load on the winch.
Of course, if the battery isn't strong, nothing is going to work for you either way.
Here's a wiring diagram as well, just in case it helps