Rob the only way I'd know "cheap" to cure this is to get some big plastic bags, the best, thickest you can afford, and pack as much stuff (parts, tools) in them as you can, then use your vacuum to get as much air out of them as possible, and seal 'em.
Oil anything and everything can.
Just about any oil, if you have access to crank drainings use that
That is one reason I'm always "railing" at these guys who want to use salamander / unvented etc gas / lp and kero heaters.......they just pump more moisture into the shop.
The only long term answer is the proper vapor barrier and insulation.
If you can box the smaller stuff up and bag it and carry some of the parts into the house somewhere.
You probably know by now, I have no shop. I've thought about getting some plastic barrels with removable lids, packing them with "stuff" and then pumping them full of CO2. Sealed with a bunch of dessicant would work, that costs money, too
Insulation in warm climates / high humidity are much different "up here" where the primary "thing" is hot and dry, or cold and either dry or wet. "Down there" you can get into cooler inside, hot outside, with a high dew point. This means that even (improperly) insulated / vapor barrier, you can have moisture problems in a building in high humidity conditions.