"We never have any problems with this sort of shit.
Damp basements, springtime warm/cold cycling....yes. Of course."
Then you obviously DO have the exact same problems. It's not springtime, but here in the south, we get the same warm/cold swings in fall and spring. Add rain with fast temp shifts and you get condensation. low country is on the southeast coast of the US. Prevailing winds are out of the northwest, so temps in Savannah are more influenced by the continental climate than the Atlantic (opposite of the west coast, where the ocean controls the climate...ie, San Francisco).
The solution is just as has been said, you need to insulate and get some form of climate control/humidity reduction in there. Before I insulated the machine shop area of my shop, I walked out one morning to find condensation from the floor and metal roof causing an indoor rainstorm on my machinery. Insulation helps wonderfully. Just slows the rate of heating and cooling to the point everything can keep up. That said, when I go out and open the door today, I'll probably get wet machines as soon as the rainy moist air hits them. I'll kick on the window unit AC/heat and dry things out in a few minutes.
Light bulbs, as mentioned, work great. I had an old fridge with a light bulb in it that I kept my micrometers, cutting tools, machine arbors, tool holders, and other critical bits in before insulation. Never a spot of rust on anything. You might also put a dehumidifier in there, once it is insulated. Sealing it up better might help some, but the root problem is temperature swings. There is virtually no way to keep the humidity out without an AC or heater along with the sealing, so your first priority needs to be insulating.