I've heard the Rem-Oil gums up like WD-40.
Wd40 doesn't gum up, but you don't want to get it on your primers.
Also I use Remoil, have for years and it is a very light oil. I have never had it gum up. Being a reloader I have shot some dirty and sooty powders too.
I have never seen a gun rust after being cared for the way you described but your problem goes beyond your guns if you have that much moisture in your safe room.
Rice will absorb the moisture well. Think volume and surface area until you find a way to dehumidify that room or move it somewhere more dry. A nice large baking pan with a inch of rice should do the trick.