Agree, as the op surmised, the problem is on the control side of the relay. Whether or not it's the signal from the switch or the ground through the case is the question.
Seems like the op is aware of voltage drop testing even if he doesn't know what it's called.

So here's how to check: take your meter,positive lead backprobed into the signal wire from the ignition switch, negative probe to battery negative terminal. What voltage do you get? If less than 10, the problem is towards the ignition switch. If more than 10 the problem is likely a poorly grounded relay.