LUBE THE SHIFTER....... Sounds like the grease has gotten old and thick. Can you MANUALLY shift the rear Der. by pulling the cable, then releasing it?hen the SHIFTER is sticking.
If lubing the shifter doesn't fix the issue, i'd suspect that the upper roller on the derailleur is heavily worn, and has way too much side-play in it's bearing/bushing.... You mentioned a bent der. hanger, and THAT can cause excessive wear in the upper roller!