Here is my attempt to shade the chairs. I spent a lot of time at it, and it ended up being not strictly dithered but not strictly segregated blotches of color either.
Overall, I am pleased with the effect (but I realize that I still have a lot to learn in 2d pixel art). I tried to simulate shadows under the seat on the legs where they meet with the seat.
I chose a black background instead of a transparent background to make it easier to see the contrast of the brown hues without bleeding or washing into the background color.