2005 Lincoln ls runs in idle but rough and when pushing gas pedle rpm drop.

SOURCE: ROUGH IDLE

fix this problem by using trial an error
u could have bad sparkplugs the smell of gas could mean bad oxygen senor an cloged fuel injectors
try changing your sparkplugs an get your injectors cleaned

SOURCE: 87 Supra idles rough and dies out

I would suspect a vacuum leak. At idle under high vacuum the air fuel mix is being leaned out until you have misfiring so bad the engine quits. The exhaust smells rich because even a lean mixture, if it doesn't light off, if full of unburned fuel. At road speeds, the vacuum is low so the engine doesn't care if there is a small leak somewhere. It is getting most of its air through the open throttle. Check all your vacuum lines but remember that even if every line is okay, it could still be leaking from an old cracked intake manifold gasket.

SOURCE: 2001 v6 escape rough idle when cold

To update my earlier repair on the 2001 6 cyl Escape: When cold, and when the ignition was turned over but no accelerator pressure given, the rough idle finally returned the P0303 code again when the check engine light came on indicating that it wasn't caused by the plugs or the coil. FYI, I had replaced the intake manifold gaskets during the original repair and just prior to that I had replaced the fuel filter which was very dirty (I cut it open to see).

The repair manual referred to fuel injectors as a possible problem. I have run three tanks of fuel using different injector cleaning products and am very pleased with the results. Cold starts are much smoother and I am unable to coax out the P0303 code or any other code.

My opinion, change the fuel filter, use a fuel injector cleaner for a couple of tanks, change the plugs if the problem still exists. The coils were not the problem on my vehicle.

SOURCE: 2003 Tiburon GT, RPM's drop way down and idles rough at stop's?

More maintenance history needed. Oil changed regularly, timing belt replaced at 60 K miles, Engine air filter & intake air hose inspection?
Causes of the problem you describe are many. Recommend you have a technician check your throttle position sensor for any bad spots in the potentiometer... most likely cause when there is NO Check Engine light.

SOURCE: idling rough. The coils and spark plugs have been

id clean the throttle body before spending more money on anything else, as carbon build up can cause rough idling.
you can clean it with throttle body cleaner, or just gasoline and a toothbrush, then wipe off the build up.