bigbassmann,
I am not familiar with a '70 C10, but I looked at your rig in the gallery. Yours appears to be a step-side, so you have individual lights bolted to the bed...right.
You said you are measuring voltage at each of the tail light sockets. Where were you grounding your meter or test light to get a ground for your reading?
What would happen if you signaled a left turn, which you said works, used your meter to confirm the blinking voltage, and NOT moving your ground, test the tail light socket...using the same ground. If it proves good, technically the lamp should work when you stick it in.
If any doubt exists as to the quality of ground for the light assembly, you can use a little patch cord clamped to a good clean spot on the frame and a good clean spot on the light.
I wonder if the little solder button for the taillight could have paint covering it. Scratch around some more with your meter.
Stuart