I recently got rid of my 2002 Taurus that was doing this very same thing, only much worse. Two days ago I let my car run completely down to E and after I filled it up I noticed it was slightly shaking at an idle. It got progressively worse throughout the day each time I drove it. Yesterday the sputtering was still there and much worse. I went somewhere and after 45 minutes when I went to restart, it almost stalled out on me and idled really low and sputtered a lot until it warmed up for a few minutes. I put some fuel injector/carb cleaner into the tank and let it run for a few minutes and then my low coolant light came on. Checked coolant level - it's normal. Temp gauge showed normal temp level. I often let my old car run down to E with the price of gas. I'm curious if it could just be junk from the bottom of the tank getting sucked into the lines and messing with sensors, etc? The low coolant light did not come on this morning. Any info you have would be helpful!
In tank fuel pumps are cooled and lubricated by the fuel in the tank. When you run a car out of gas or very low, it pumps a lot of air through the pump which overheats it and causes some severe wear and many times, failure when they cool off.
You will need to use a fuel pressure gauge to verify this