What is your make of water heater? Atwood or Suburban?
Atwoods have two small devices that open the circuit. The thermostat and then the ECO. They also have a thermal fuse normally hidden behind some heat shrink tubing adjacent to the thermostats.
The thermal cutout opens as a safety switch to prevent damage.
The ECO or the thermal fuse can shut down the electric going to the circuit board.
Both the thermostat and the ECO can be tested. You could wait for the unit to go into failure, test the thermostat and the ECO. The thermostat should not have continuity. The ECO should have continuity under normal operating temperatures.
Dripping is normal. Not a real problem. I would get the unit operational, then think about replacing it. They are a standard relief valve available at most hardware stores. Just need to get one matching the name plate disc on your existing one.
