First about the oil - food grade mineral oil will work as well as any other, and costs maybe 1/100 of the specialty stuff. As somebody already said though, it needs to be dry first - a minute with a hair dryer after wiping it down well should do. I haven't had a problem in SE Michigan but spring has sprung, so it will get a bit more humid. A good drying and the silicone sleeve should work just fine without any oil I think.

I've tried at least three ways to wipe the blade, water (from my DE habits), washcloth that starts dry and gets wet, and the sponge. Not sure it matters really, presently I'm rinsing with hot water and giving a quick wipe with the shave towel on the counter. The sponge was to me a pita, but it worked as well.

YMMV