2004 f150 4.6lt Triton

Sounds like your cylinders are misfiring. Make sure that the plugs you installed are to spec, meaning make sure they were gapped correctly before installation and that the plugs were made for that particular engine. If coil packs were replaced, those should be fine.. You claim to have cleaned the fuel rails but you could have a failing fuel injector. For starters, I'd get the OBD-II working to check out the codes and then just google for code interpretations. Hope this helps.

Since you spent all the cash I suggest you spend another hundred bucks and have a mechanic do a live data analysis on the engine while it is idling. It does sound like a misfire in one or more cylinders, but what is causing the misfire is the key. It could be bad injector(s) or loss of compression in one or more cylinder. The live data will indicate which cylinders are misfiring narrowing the cause of the problem significantly.