I use a chux with some oil on it, and wipe it on the blade much like you would double over a tissue to wipe both sides of a lense on a pair of sunglasses or whatever. Always be careful of the edge of course.

This is also how I dry my razors using a towel or, when I am lazy, the front of my tee shirt. Many of my tee shirts have tiny little cuts in the front of them where the edge has just grazed the material as I wipe.

James.