So, after months of searching, testing parts, replacing rubber connectors, checking lines, swapping filler necks, and buying a new gas cap, I have finally narrowed the problem down to the gas tank. I had my Dad help me drop it today to see what we could find.
We had some trouble getting the straps off; even had to saw through one of them. Lowered the exhaust by unbolting the muffler hangers (if you don't move the pipe, it makes it really hard to get the tank out properly).
Here it is, our culprit. The fuel tank pressure sensor. The "head" is snapped right off. I had to use some pliers to pull it out of the hole in the tank. It looks like someone actually tried to glue it back together after it broke. I also looked at the date imprinted on the bottom and discovered that it'* the OE one from the factory.
We are convinced that someone (a prior owner) had serviced the tank before, and did a poor job of putting it all back together. My theory is that the sensor somehow got broken while they were working on the tank and they just never replaced it.
And while I'm down there... I decided to replace the fuel sender lock ring and fuel pump strainer. Old vs. new in the next pic.
I'll be putting it all back together in the morning. Next up on the list is a rear brake overhaul with new drums, shoes, wheel cylinders and springs.