Headlights make engine run rough

One nice thing about a relay set up is that if one should fail (something that's never happened to me in 25 years of running similar set ups in other cars and motorcycles), you can simply unplug them and plug you original back in.

Although they are not complicated to fabricate, there are ready-made plug-and-play solutions out there for our trucks that have quality materials likely far less expensive that might be sourced for a oneoff project.

Went and voltage drop tested the headlight grounds:
I found between 0.08 volts and 0.25 volts of voltage drop between the headlight ground pins and the battery terminal, with headlights and high beams running and engine off.
On the passenger side, I found that that voltage drop was almost halved if I measured the drop from the pin to the fender rather than from the pin to the battery.

So it seems that my headlight and chassis ground cables are all only a little iffy. I might try to clean the contact surfaces a little, but it seems my problem would be fixed more if I did the relay conversion since my voltage drop was below the general 0.4 volts and only slightly above your 0.2 volts spec.

A relay on the +12V side won't have any effect on the voltage drop on the ground side.  To fix that, simply run Copper wire from the factory ground point directly to the battery (-) terminal, or one of the other terminals on that cable.

can0fspam - Yes, you should work on the ground problem.  It isn't bad, but it isn't good.  And as Steve suggested, a wire from the headlight to the battery ground.

But, if you buy a headlight relay harness, it should have a ground wire that will give you that opportunity.  So instead of fixing the ground issue now, I'd go for the harness and fix it then.

And this ground issue is why in later models Ford added a bunch of ground wires.  Using the body's sheet metal and the fasteners that hold those pieces together as the ground path was problematic.  It was probably adequate the day the truck rolled off the assembly line, but started downhill from that point.