1996 Ford Thunderbird EGR Valve/Temp. Gage

Engine Performance problem
1996 Ford Thunderbird V8 Two Wheel Drive Automatic 107k miles

I own a 1996 Thunderbird LX Sport with the 4.6 engine. Recently, I filled the tank to about 1/2 and then drove on the expressway for about 40 minutes. Car was running fine until I shut it off and then started it after it sat for about 10 minutes. Then, the car began to idle rough and the check engine light illuminated. I had the codes scanned and it said "faulty EGR Valve". As of today, the check engine light is off and the car idles smoother. I also noticed the tempaerature gage does not return to the C (COLD) mark on the gage, almost as if the engine is not cooling efficiently. However, the gage returns to C after the car starts up and runs normally. Could this be the cause of the faulty EGR? Does the engine run hotter when these valves go bad? Thanks for your time!