The Choctaws of Skullyville who were Ainsworth's are documented, but the Choctaw lineage does not come from the Ainsworth family.
David Ainsworth born 1785 married I think it was Martha Spencer. Two of their children were David Spencer Ainsworth, he never married a choctaw but his children did, and Thomas Garret Ainsworth. Thomas married Martha Brashears, who was part choctaw being the daughter of Delilah Juzan (my relative) and Jesse Brashears.
Both brother's moved to Oklahoma territory when it was part of the Choctaw nation from Mississippi, having been I believe in Washington county territory first Alabama. I believe there father was Levin Hainsworth listed in 1803 property list, but have no documentation to prove it.
By the way, Delilah Juzan's mother is the niece of Pushmataha, and it can be documented. As a descendant of the same woman, not so happy with the South Carolina folks who everyone is saying is related. Pushmataha didn't live in South Carolina, and there is nothing mentioning the chief Phillemaya or a Mataha saying he did either. And he didn't sign the treaty of Hopewell either.