+1 on WD-40 to clean off the old grease. It's cheap
and the spray straw really gets into the crevices. Get the big "40% extra for free" can at HD.
There was a time when men-were-men that boats heeled more and sheet winches spent a fair bit of time submerged along with the leeward rail. Those Lewmar
winch instruction came from those days (and a lawyer)
The winches on my current
boat get an occasional bit of spray but not the sort of salt water
firehose that gets up inside the gears.
And the winch greases are much longer lasting. Use the grease recommended by the winch manufacturer. (and only light oil like 3-1 oil on the spring pawls)
I only grease my winches every 2-3 years and most still look perfect when I lift
the drum off for the cleaning
This infrequent greasing also means that I have far fewer opportunities to drop bits of the winch overboard
Here's an old trick, before taking the winch apart, cut a hole the size of the winch in the bottom of a cardboard box. Place the box over the winch for both assembly and dis-assembly. This will give you a fighting chance to hold onto all the pieces. Also, take cell phone
pictures at every step of dis-assembly to get things back together again later.