They are thermostatically controlled. The tank should still have water but at a less than full level. Next time turn the WH control to pilot.

Tightening the stem packing nut or adding additional stem packing material will have absolutely no effect on stopping a leak where the water is leaking through the valve. The stem packing is solely to prevent water from leaking around the stem shaft. To answer the original question, "Is there a packing washer inside the valve?", the answer is "Maybe, but it is more likely that the valve is just a cast plastic body with a plastic internal seat and the stem is a one piece plastic casting including the washer face. Back in the early 80's some engineering genius sold the water heater industry a load of baloney about how much heat energy was being radiated through a metal valve and encouraged the use of the plastic valves as an energy conservation measure. Of course the water heater manufacturers liked the idea because the plastic valves are about $2 cheaper than a brass valve, and when you are making a million units a year, saving $2 a unit adds up to a substantial savings. On the other hand, out here in the real world it has been shown time and time again that the first time that valve is opened it is anybodies guess if it will ever close properly again. The simple solution is to drain the water heater and remove the plastic valve then replace it with a brass "3/4" Boiler stop cock valve". Some of the high efficiency water heaters have a thicker insulation blanket so it is sometimes difficult to install a common boiler cock valve. In this case there are two solutions. 1.You can install a short 3/4" pipe nipple into the tank, then install a female "boiler cock valve" of they also make a "long shank boiler cock" that has a longer male pipe end on the input side of the valve. The replacement valves can be purchased at any local hardware or home supply store for about $8 to $10. You will also need either some pipe dope or Yellow Triple Density PTFE (Teflon) tape" to put on the threads before installing the valve.

Replace the WH... no question about it. Make sure your softener is giving you 0 hardness water even at peak flow. If you're paying for soft water you ought to get it all the time.

@ Vith... "If the pressure of the water in the water heater (and the water heater cold inlet and expansion tank) is higher than that of the city then you will get flow from the water heater into the cold distribution till the pressure equals out". Not really... in a closed plumbing system with a thermal expansion tank the tank will accommodate the water volume increase when the pressure rises just a couple PSI over the static pressure. In a closed plumbing system the increased volume of heated water has only two places to go as long as no one opens a faucet. One place is the T&P valve and the other is to force open a toilet ballcock valve. Every other valve in the plumbing system is a positive off valve. "I would rather ensure no thermally expanded heated water can ever be capable of backing up into the cold line even though I have an expansion tank. That is just my opinion". You're welcome to your opinion but you're worrying about something that doesn't occur. We're not looking at gallons of water moving in the system when the water is heated. When thermal expansion burps a T&P valve you usually get an ounce of water while the pressure is relieved. Much more than an ounce moves into the thermal expansion tank as the water volume rises when the water is heated but if the tank is correctly sized and the air charge correctly set you will never get anything but cold water in the expansion tank. If it makes you feel better you can install a dozen check valves and all that accomplishes are more connections for a leak to occur and more moving parts to fail.