My 03 Lincoln LS started idling rough only in the mornings when first started. Rest of the day it starts and runs smooth.

Tune up items, dirty IAC, Dirty EGR, injectors dirty.

PROBABLY NEEDS A TUNE UP AND NEW VALVE COVER GASKETS, SOUNDS LIKE OIL IS LEAKING INTO THE SPARK PLUG TUBES AND CAUSING THE MISFIRE

ck for poss.vacuum leak at upper to lower intake manifold gaskets

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.

replace the fuel filter and was your fuel pump from autozone if so there is your problem

run the fault codes first
use a quality injector cleaner to clean up the injectors

Just figure i would try and help, Did you get gas just before this happened? It could be as simple as bad gas, However i have seen problems with them having bad ignition coils.
I think its like $300 for a diagnostics at a Lincoln dealer and i know you can get a set of six ignition coils for $225.00
Well good luck in the morning and i hope it is just bad gas, Well let me know and good luck.

sticking valves. the short run of the engine free's them enough to enable runnig

Sound like a bad ground wire going from engine to firewall...