I would expect SUPPLY voltage to drop when the key is off. I would not expect the A/C control to drop voltage itself. The two other posters who've tried to help obviously believe the same...in providing suggestions where to look. The mere presence of a relay on that circuit implies [high amperage] voltage is supplied to the heater/controls when the car is turned on. That's why relays are used...to transfer the load "around" the ignition circuit.

Check other components on that circuit to see if they are still "live". How long has it been that way? Any chance it's been rewired incorrectly?

Did you try swapping relays (same ratings) to see what happened?