The inside switch for turning the WH on electric is 12 vdc that operates a control relay to supply 120 vac to the WH circuitry and heating element. I believe that control relay is the only element common to both 12 vdc and 120 vac. Relay is typically mounted to the side or back of the water heater.
But the problem may very well be that the circuit downstream of the switch is shorting out. Possible the wire to the relay has been damaged and touching ground somewhere between the switch and the control relay. The A/C thermostat uses 12 vdc. And if the fuse that supplies power to the WH control switch also supplies power to the rear A/C thermostat that would explain what is happening when the fuse blows.
The upper diagram shows the 12 vdc to operate the relay. The lower diagram shows the relay supplying 120 vac to the small black switch on the water heater.