Hot water heater is blowing the relief valve...

relief valve does not normally drip

So the relief valve works on pressure or temp or both?

T&P means temp and pressure?

The unit is an Atwood G6A-6E.

So when you say the eco has continuity, you mean zero ohms below like 180?
And the thermostat is open circuit below 130 deg?

T&P valves are easy and cheap to replace. I'd start there.

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.

RV water heaters use 2 t-stats
One is the 'normal' temp t-stat. It is what allows heating & reheating (after cool down) of water.
Suburban uses 130*F fixed temp t-stat
Atwood uses 140*F fixed temp t-stat

Other one is the ECO (Energy Cut Off) t-stat (High temp shut down). If normal t-stat should fail the ECO shuts down heating and 'locks' out further operation until 'fault' has been reset.
Suburban uses 170*F fixed temp t-stat and has a reset button that allows t-stat to be manually reset
Atwood uses 180*F fixed temp t-stat that will auto reset when water temp cools off enough
Both have 'fault' lights and that has to be reset (Turn switch OFF then back ON)

Atwood also has a thermal cutoff fuse.........it is a one shot device that trips at 190*F due to blow back out of combustion chamber. When it trips it shuts down all operation and device has to be replaced.
It is connected to normal t-stat (has a clear tube over fuse)

Weeping T&P Relief Valve can be caused by:
*Weak T&P .........replace
*Loss of 'Air Pocket'
Water heater is designed with an air gap at top of tank. Air can be compressed so as water swells from being heated the air allows for this increase w/o a big increase in pressure.
Lose the air pocket and pressure increases as water swells from heating causing T&P to weep.
Establish air pocket
1) Turn off water supply
2) Open faucet hotside to relief pressure
3) Open T&P via Lever....hold open ntil water stops running out then let valve snap close
4) Turn on water supply
5) Close faucet
*Dirty T&P
Open via lever let snap close couple times to clear valve seat (CAUTION: Do with water supply on so water blows out----water can be very HOT)
Then re-establish air pocket


Which Brand/Model do you have and if Atwood which year (Changes made in 2004)

So you are not thinking the thermostat?

What does a thermal cut off do?
Is that what cuts off the propane?

Dripping from TP valve is not uncommon, especially if it not seated well enough. Try opening and closing it a couple of times. If that does not work new valve from your local big box store will fix it.