I have a lincoln ls v6 2004 the check engine light is on and the and the shifting seems rough

Do you have a check engine light flashing when it runs rough?

Probably problem with the (ETC) electronic throttle control system . If the engine light is lit ,didn't anyone check for DTC'S - diagnostic trouble codes ?

Do you have a block heater? You can have on installed if not, then plug your car in over night. You car will run much smoother in cold weather.

perform compression test

The trans is shifting rough due to the engine issue. Is CEL light on? DTC codes?

Check Engine Light flashing or solid on?

tune up ? emission control component problem ? Go to autozone they will scan it for free.

Need to pull codes. Give me the codes and is it a V6 or V8

You can get an ACTRON scanner from Sears. The regular or plus version ($179). Harbor Freight sells scanners too.
But, you can go to AUtozone or most other stores and get the codes read for free when the CEL is on.
Get codes, report back before buying anything.
Sometimes codes are caused by other engine problems that influence the sensors and turn the light on.

The rough idle will usually be a misfire code.

I checked for TSBs. None found for 3.9L engine

sounds like tranny computer is bad take it to a grage and have a computer put on it cost about $50.00 to do this

mileage on vehicle?...could be clogged catalytic converter