Having the Diagnostic Trouble Codes (DTC) codes read that triggered the Check Engine Light is the first thing to do!
Those codes are likely going to lead to what is causing the "jerking" problem
Once you have the codes, do some intensive research into what triggers the codes - do NOT blame the sensor and replace it without good cause.
I own a 2003 Taurus, and had the same jerking and sputtering, with the light on. What I found was the codes pointing to misfiring on some cylinders. For me, I had to burn out all the remaining gas in the tank till the gas light came on. Then added a bottle of fuel injector cleaner to a fresh tank of pure gas. Not the gas with the corn syrup in it (ethanol) but real, pure gas. Some places still sell it. Turns out the corn-gas was choking up the injectors. Many older cars (pre-2006/7) do not digest corn-gas, and as a result, using too much of it will clog things up. After doing that, and resetting the car's computer, the light never came on again. And since I kept using real gas instead of the corn stuff, I never had the jerking and sputtering since.
See "Related Questions" below for more
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The 2003 Ford Taurus has a 3.0 L base engine size.
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The 2003 Ford Taurus's engine produces 155 hp @ 4900 rpm.
The 2003 Ford Taurus's engine produces 185 ft-lbs. @ 3950 rpm.