Thanks for the reply. Yup, at some point the wiring was "modified". There are a lot of non-factory splices in the wires that run from the engine compartment into the cab. It looks like there was once a "tee" in the heavy red wire that comes into the cab from the engine compartment...and that "tee" burned off. There was also a "jumper wire" from one part of the fuse block to another that looked professionally done, but didn't look "factory" in that the terminal ends didn't have the plastic insulators over them like the rest of the wiring does. Apparently the jumper wire was providing power to the part of the fuse block that was supposed to have constant power, and a wire that provided "switched" power to the rest of the fuse block was removed.
Actually, the whole wiring system did not turn out to be the nightmare I expected. When I pulled the truck out of the desert about 80 miles from here and got it home, hardly any of the lights worked...no tail lights, no brake lights, erratic turn signals...along with the aformentioned problems.
Luckily, I methodically checked everything out (unusual for me!) before I started cutting and splicing, and it turned out there were several simple reasons for the problems...bad grounds in one tail-light, a bad turn signal flasher unit, a mal-adjusted brake light switch..etc.
Now everything works, but I still cannot figure out why they had the high-beams headlights pointed practically straight at the ground...