Will, I haven't had this problem with the shade screws but have had it with other areas. I tried a number of anchors but found two things that seem to work best for me.
One is fill the hole with elmer's wood glue then break off as many toothpicks in the hole as you can. Then when it's dried, reinsert the screw.
The other is to use an anchor. I've tried several but find this one, an "auger anchor" works far better than others, short of an actual toggle bolt.
You use a screwdriver to screw this anchor into the hole, then the screw screws into the anchor.
I've used these anchors to hold a magazine rack to the wall in the water closet, a four hook clothes rack on the wall in the bedroom, and a paper towel holder to the thin paneling on the island in the kitchen. All have continued to hold very well for the past six months or so we've had this rig.
I can't guarantee it will work on the shade but it has worked well on other things for us.