Ohhh yeah that thing is nuts. Almost never accurate and sometimes just moves to empty right before my eyes. It hasn't been right since I got the car.
Actually that could be the "standard" behavior on the Taurus. There is some "order" to these gauge movements. If a car has been driving on a level surface for some time, the gauge should be at the "correct" position. When the car is driving downhill for some time, the gauge moves up. If the car is has been driving uphill (or was parked facing uphill), then the gauge will move down. On level position marks are separated by about 3 gallons. If gauge is at the top mark, the gas tank is full. The next mark, about 13, followed by 10, 7, and 4 gallons. The lowest mark corresponds to roughly 4 gallons. When the gauge hits that position, the gas light comes on and stays on. Unfortunately, the behavior means that if say you have 7 gallons and drive uphill for some time, the gauge will go down to the 4 gallon mark, and the gas light comes up and usually stays on even when car is driving on a level surface. If you observe something different, then there could be a problem.
Wait.... do you have the 'u' code engine or the '2' code engine? 8th place in your VIN.
If you have the '2' code engine, you have the flex fuel Vulcan. Those engines are known to have a faulty module behind the radio. It does something with measuring the ethanol versus gasoline or something: Fuel Gauge Erratic - 2000 Flex Fuel 3.0
-'66 Ford Country Squire wagon 390/C6
-'87 & '89 Mercury Colony Park wagons 302 EFI/AOD
-'90 Lincoln Town Car 302 EFI/AOD
-'03 Lincoln Town Car 4.6L/4R70W