Another vote for coil pack. Aftermarket ones work better than Ford's own. At least from my Ford Expedition 5.4 experience.

I had the same problem on an '02 4.6. It was one coil pack, then another. Ended up just replacing them all.

A few thousand miles later the oil pressure went to zero then boom.

COP. Feels like a misfire or a tranny shudder. But the CE light should be coming on...

obd2 laptop software - Google Search
Some are free and some close to it.
Worth a shot.
MIL light should illuminate for a second with key on engine off, unless burnt out.

Check all big connections of course and wiggle while running.

If looking for software make sure that the "advanced diagnostics" are included otherwise you might as well go to Pep Boys or AutoZone and all you'll get are generic codes and some donot allow you to clear codes. These modules are generally $50 and up for each one. The ability for monitoring when driving is important especially if you have a problem that comes and goes. Also ask how they handle laptop upgrades and if you can move the software to another computer if yours breaks. As with most things, you'll get what you pay for.

As a side note, about a year ago I wanted to get my wife's Caddy checked for a SES light and all 4 auto parts stores said they were not allowed to do that any longer under CA law? Said you had to have a "repair" license and not just sell parts?

+1. Same situation on my son's Y2K F-150 with the 5.4. Bad coil pack. Not too tough to R&R. Ran fine afterward.

I recall that engine had a composite intake manifold (not metal)? Lots o'problems with that, and I think they went back to metal versions in later years.

The plastic intake manifolds will leak anti freeze , that will then weep down into the spark plug wells and cause missfires as well. I tried to re-gasket a few, but they still leaked. I just replace them when I see them leaking coolant now.