Water Heaters - Bad thermostat?

I have a 50 gallon Select State gas water heater that is giving me problems. The hot water temperature is extremely inconsistent. Sometimes the water is barely lukewarm and sometimes when I turn on the shower it is scalding hot. Also, we generally only get enough hot water for 1 shower at a time before we run out.

Last night the temperature/pressure relief value was tripped and very hot water was flowing out of the valve at a regular rate. I noticed the burner was still firing! I had to turn it all the way down just to get it to shut off.

I've flushed the hot water heater and no sediment appears to be in the tank. That's hard for me to believe as well since the tank is nearly 9 years old.

Does this sound like a bad thermostat? Should I just replace the thermostat or should I replace the entire tank considering its age?

I'd seriously consider replacing the entire water heater. Nine years is at the upper range of the average life of a gas-fired water heater and replacing just the thermostatic gas valve could be sending good money after bad.