Can you get underneath the sink and unscrew the pipes to take it apart and clean it out manually? There probably a horizontal pipe between the two sinks, and that is what is clogged. Be sure to
a) wear long sleeves (some garment you don't care about) and rubber gloves - there's Draino in there which is caustic, its whole role in life is to eat through things, so you do not want it on your skin!
b) clean everything out underneath there and have a bucket poitioned for the water
c) open windows before you start so the room is well-ventilated ...