If you put your HWT on bypass by shutting off the inlet to the HWT and opening the valve between the inlet and outlet of the HWT(as I think you may have done) then cold water should run out the taps when set to hot (on it's own) and cold (on it's own). If that is good, then shut off the water pressure in, release the pressure, and pull the plug to the HWT and drain off all the water. Now, you are in bypass still, put inlet water pressure on again with all taps in the off position. If water comes out the HWT it is getting in by a partly open hot water line check valve. Not the relief valve, but the inline check valve. It could well be frozen partly open.
If nothing comes out the HWT then check the inlet line to the tank. To check the inlet to the tank, go off bypass and drain the tank by pulling out the plug. Make sure all taps are in the off position. Connect to water. Water should flow into the tank, and onto the ground through the pulled plug. If you seem to have a good flow out the drain plug hole, put the plug back in and turn on a tap when you are not on bypass. The tank should fill and only release air out the taps until the tank is full, then water should come out the tap. If it's slow, you have to change the check valve on the outlet of the HWT. This is not the relief valve on the top of the tank, it's the check valve.
The check valve is only about 3/4 of an inch long, and until I had a problem, I didn't know it existed either. Compare the cold water inlet line, and the hot water outlet line and you will see the check valve in the line.
Mike
2012 Winnebago Impulse Silver 26QP
2005 16.6 Double Eagle
2018 Jeep Wrangler JK
previously Snowbird Campers,
Triple E Motorhome and Fifth Wheel