For regular day to day cleaning I like to use a sticky lint roller or the small brush attachment to the vacuum to remove surface dust.
This will eliminate smudges and stains than could come from brushing the shade.
Most cotton lampshades are washable, the trick is to wash it evenly. Fill a bathtub or sink basin that is deep enough to completely submerge the shade with lukewarm water and a squirt of baby shampoo.
Holding the metal support, dip the shade completely into the water and gently clean the entire surface with a soft cloth.
Empty the tub and refill it with clean water then dip the shade in several time straight up and down to rinse the soap and dirt from the surface.
Set the shade on a clean, white towel to dry. That's all there is to it so stop turning that shade around trying to hide dirt and smudges! I'm Mrs. Fixit and it's just that simple!