Yes we are distant cousins.
My ggg grand parents are Thomas Fields and Nannie Downing. In 1839 they were with the east Texas Cherokees when the Texas army attacked and pushed the Cherokees north to Indian Territory, as you probably already know. There were a lot of Cherokees killed including Thomas and Nannie. From the information that I have, I am about 95% sure of this information. Thomas Fields is the youngest brother of your gggg grandfather George Fields and Chief Richard Fields. "Small world."
Thomas and Nannie's son, Richard F. Fields b. 1823, d. 1879 is my gg grandfather. I had thought for a few years that the Richard in the 1866 photo was my gg grandfather until I got the Civil War and prison records of Captain Richard Fields. The ages were differant and that's when I realized that this was Richard George Fields b. 1808, d. 1873. It is easy to get these two Richards mixed up. They both were very educated. Richard F. was a lawyer and Cherokee Supreme Court Justice and very involved in Cherokee politics.