From your limited description, I also do not believe your dimmer packs are defective. The mains ground wire (green or yellow/green) SHOULD be attached to the chassis, that's what it's for... to attach the metal chassis to earth ground so a short to the chassis will follow that path rather than thru YOU.
You cannot check continuity (ohms) with the dimmer pack powered on. Are you checking the DMX line or the AC line?
As Sandy says, if you could provide the exact manufacturer and model number, you will get more responses to your question.
Normally the problems you describe are DMX cable problems like shorts, swapped wires, too many daisy chains, etc.
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