Definitely time to lose the cable.
As for the other issue, it sounds like the winch motor isn't getting grounded properly. This happens a lot after a winch is refurbished (the fresh paint is enough to block the ground) and I've seen it happen more than once on winches that have been working fine for a long time. The best solution is to weld a post to the motor case and attach a ground directly to it. Most older winches have a ground wire that runs to the bolts that hold the motor to the winch (and the end plate cover on). Since your solenoids typically also have a ground running to that location, the solenoid will operate; however, the ground has to go through the thin metal of the end plate to get to the motor for it's ground.
For a quick fix, you can remove the end plate, scuff up the area where it mates to the motor, and make sure that the ground is getting good contact with the bolt/end plate. As I said before, a better option is welding a post onto the motor casing (I normally just take a decent bolt, grind the head into a nice round base (take the corners off), put a sacrificial nut on to protect the threads from spatter, and weld it to the casing. Once you have it in place, you can back the nut off to the height of the other posts and trim it off flush with the top of the bolt, run the bolt back down, chamfer the edges, and back the nut completely off to clean up the ends of the threads.
After paint, it looks like it came from the factory that way and you should never have a ground problem again.
ďThe test of the machine is the satisfaction it gives you. There isn't any other test. If the machine produces tranquility it's right. If it disturbs you it's wrong until either the machine or your mind is changed.Ē ― Robert M. Pirsig, Zen And The Art Of Motorcycle Maintenance: An Inquiry Into Values