If turning off the power that controls the contactor (i.e., it's coil control) de-energizes the relay, then, yes, you have a problem in the switching circuit. As to how to isolate it to a particular switch, that could get harder. First thing I'd probably do is flip each one and see if they 'feel' proper. On older switches, sometimes you can feel or hear that they are not switching properly.
I'd consider updating the switches to something from X10, or Insteon - i.e., one of the remote controlled type of switch. Also, Lutron's Maestro switches (I know there are dimmers, not sure if they make on/off in this series), uses a much simplified multi-switch wiring, and might be a replacement possibility. Those can handle up to 8 switches controlling the lights with just 12/3 wire running between them. It uses the third wire as a trigger that is tied to all of them in parallel. One master switch and up to 7 slaves.