I've run into the same problem (humidity) with both rifles & shotguns.
For 12ga. shotguns, I use a 10ga brush as a chamber brush, and for rifles, such as the .308 & .30-06 family of cartridges, a .50 cal bore brush as a chamber brush
( those cartridges have a 0.473" base diameter, so the 50 cal brush works better than a .45 cal brush - "Pro-Shot" [cleaning rods & accessories] also makes a specific chamber cleaning brush ). When things get bad on the shotgun, I have taken an old aluminum Hoppes cleaning rod front section and chucked it in a 3/8 drill. Slipped a piece of clear vinyl hose (available Home Depot) rover the rod to prevent it fom touching the chamber, and gone at it with the "chamber brush & Hoppe's No. 9 - after wrapping a little 0000 Steel wool around the chamber brush.
Does a great job ! Same process for a bolt rifle - only with a sacrificed fron part of a rifle cleaning rod chucked in the drill. Neither the brush nor the super-fine steel wool hurts a thing.
Leaving the bolt out of the rifle (after a thorough cleaning & oiling - store muzzle down ) seems to help too. For shotguns, take the barrel off the gun - try and store so air can circulate. A big bag of desicant will also help in your storage area. For longer term storage "RIG" seems the best rust preventative... but don't forget to clean it out before you use it !!!!