Over the decades I've had the best results using a soft, damp (not wet, just damp) cotton cloth like an old tee shirt (I use cotton cloth baby diapers... absorbent, soft, thick... perfect) to wipe off skin oil, smudges and grim. Then, use a dry cotton cloth to wipe off any excess moisture and buff. Include the strings, neck and body in a wipe down. Be sure to wipe down the strings; they are the biggest collector of skin oil.

Some other items to consider:

You write there is a little rust on the pickup poles. Depending on how much rust, its likely to alter the performance of the pickup. You might want to address this at some point, but since its not a primary instrument, probably not a major item... as long as there is no rust on the pickup windings or any other wiring and electrical parts.

Also, given the environment I'd suggest changing the strings more often that one would otherwise.

Keep an eye on those tuners because if they rust out too much you won't be able to get in tune or stay in tune. But you say you've replaced those (most likely with covered tuners.. the kind where the gears are not exposed) so you have that taken care of.

Playing at a beach resort outdoors I would put the guitar in its case or gig bag when not using it. If not in use, protect it from the elements, which you probably are already doing.

Its a tough environment near the beach and last thing you need to happen is for the guitar or amp fail while you are playing. Its not a pleasant experience I can tell you.