The part on a bike that is damaged in nearly every crash is the derailer hanger. That's the little removable bit of aluminum that attaches to the frame and serves as a mount point for a derailer. Because the part of the derailer that sticks out the most from the bike is actually quite strong, the hanger always bends before the derailer (which is why the hanger is replacable). A bent hanger can cause all manner of drivetrain crunchiness, such as what you're experiencing.
The good news? A hanger is a $20 replacable part (plus labor, if you have a shop do it). They can usually be straightened as well, probably for less.