2004 Dodge Stratus Possible Alternator?

My car would not start, made the first "turning over" sound and then just made fast clicking sounds when the key was turned to start.

I was still receiving power to the car (lights, radio, etc), but I went ahead and replaced the battery.

After replacing the battery, the car did in fact start right up, however, when I come to a stop or take my foot off of the gas, it dies. I had to drive to work with a continual flow of gas going into the car, but my foot on the brake when stopped.

When the car would die a stop, I never lost "power" inside the car. When I would restart the car, all my radio/clock settings were still the same, so that lead me to believe it wasn't the alternator, but something else?

Please advise if you can, and thank you.