Fords must have EO spec spark plugs. The fact that they were gone on one bank tells me that they were not OE spec plugs. What did they replace the plugs with?

One part that affects the way the vehicle runs at temperature is the coolant temp sender. It tells the computer how hot the engine is so that it can adjust. If that sensor goes bad, the truck will run funny.

It could be a fuel pump problem. What diagnostics have been done on this truck, or have you hired a bunch of parts swappers. Every part on this truck can be tested. The idea that it is a cat problem is bogus. If you had a plugged cat, it would be almost immediately obvious after startup, not after it warmed up. Besides, cat problems are the symptoms of other problems anyway.

You mentioned plug wires and distributor? You must MUST use Motorcraft wires in these trucks. And there is no distributor. Factory plugs and wires only, all others will cause problems with your trucks powerful waste spark system.