I am in a similar situation right now.
I am moving from two locations totaling over 3000 sq feet with no garage to one location over 2000 square feet but with a garage.
We have no cold in Florida, but we do have humidity and the new location is at the Inter-coastal waterway, so there is a high salt content in that humidity
My presses and dies will be stored indoors, but the projectiles and brass has to go in the garage.
I have always put my tumbled brass in large plastic pretzel/candy/snack jars like the kind you get at the Wholesale Clubs, and I have LOTS of brass. Probably 15+ of these jars
These will go on large storage racking and will have no problems with the humidity
I also have LOTS of projectiles on hand. Here is one of the two small racks in my old location
For these, I purchased many 50 caliber ammo cans. These will keep the moisture out of the projectiles way. These are also going to be stored on the racking in the garage.
BTW, a 50 cal can with 4000, 357 Magnum JHP projectiles in it weighs over 70 pounds