I replaced both my Up and Down solenoids when I bought my boat as they we jammed very much as yours was, in preparing to do the job I read up as much as possible on other peoples experiences. Nearly all were 'I went to use my anchor winch one day and found it wouldn't work because the solenoid had seized'. One story however had the solenoid seize while the anchor winch was in use! A husband and wife were in a busy anchorage when they decided to head back to their home berth, the wife took the controls on the flybridge while the husband went on the foredeck to lift the anchor. All went well until the anchor stowed in the bow roller and the husband let go of the Up switch. The solenoid was stuck open so the winch kept going, the anchor being stowed however left the winch not winding but making horrible grinding noises and then apparently burst into flames. There was no quick way to turn it off so the husband ran back to the engine room and turned both battery switches off. This stopped the winch but also stopped the (electronic) engines leaving the wife screaming with no control in the busy anchorage! The husband tested both battery switches to see which the winch was connected to, he then started the other engine to give his wife some control and then dealt with the burning winch. Presumably he then harder job of convincing his wife that boating was still fun!
I've always been glad my solenoids seized while not in use.
could you relocate the solenoids to a dry place? mine are, had a bit of arcing issue, dismantled, cleaned, working fine now for 7 seasons. But mine were the Quick two throw ones oblong rectactular ones.