Thanks for all the insight, I thought I would document the fix in detail in case someone else has this problem. My crack mechanic got it fixed in less than an hour. To recap, the right side would only work intermittently, with no flashing on dash or either front or back light. Sometimes the back right light and dash light would flash quickly, as if there was a burned out bulb. The left side would work most of the time, but sometimes not. No problem with bulbs or fuse. Contributors from the other site were right, there is no replaceable relay. No relays in the headlight bucket. We took the switch apart and it was clean and functional, but not getting power to one side. Mechanic thought it was a ground issue somewhere. Eventually he focused on the front right indicator light. Turns out that the metal piece on the top of the light socket, which seems like it is there to keep the light snug, also serves as the ground. It had gotten bent and was touching the hot wire creating a direct short. Since this was happening, it must have been tripping a circuit breaker inside the instrument panel, helping to explain why the left side indicators were sometimes not functioning, as it thought the right one was flashing. He was able to bend it back to where it should have been, and all works fine now.
2 major things I learned about indicator lights on Thunderbirds.... One is that the clips on the indicator lens covers are very fragile. I broke a clip off one trying to put it back on, it still works but is kind of loose. Another broke a few clips right away, it will not function. I have ordered 4 replacements to have a few spares, about $10 each US. The second thing, when installing a new bulb, be very careful of the metal half sleeve/clip that you will see on the top side of the socket. You need to point the front of the bulb up so the base of the bulb goes in under the clip, then level it and twist it to engage. If you force it, or that sleeve gets pushed or bent out of position you might end up with this same issue.
Thanks guys for the suggestions, Deacon you were on the right path, special thanks to my great independent mechanic Chris Smith of Smith Cycle in Dubuque, Iowa.