Your first idea is the best one, but may be more
work than it's worth. If the
centerboard is from 1982 and only has surface rust, you may get by with simply
sanding that off and going with Rustoleum. Try it for a season and see what happens. If the rust gets a lot worse you can give it the heavy-duty treatment. If not, you'll have more time for sailing.
You do not want to leave a boat with the
centerboard down on a
mooring. It prevents the boat from swinging into the
wind, and can lead to its capsizing. I have seen this happen with Lightnings, which are bigger and heavier than your Javelin. (In Larchmont, btw, just across the
water from you.) If the centerboard is left up when the boat is moored, it won't be sitting in the
water all the time, so may rust less.