According to a book digitized at Ancestry called "The Ancestry and Posterity of Matthew, John, Daniel, and Samuel Lyle, Pioneer Settlers in Virginia," John and Susan were married in 1852; they were the parents of David and Martha. By 1860 the family was in Texas and David, listed as an orphan, was living with his uncle David and his family while Martha was living with a couple named Featherston. This suggests that both their parents had died before 1860. I don't see anything to suggest that Susan Bird had any Cherokee connection; there is no Cherokee "Bird" family in either Tennessee or Texas in 1850.
There was apparently a Dunwoody family Bible belonging to Patrick Dunwoody which went to Texas with son David and remained there. It might have a clue if you can locate it. See article in the Fort Worth Star-Telegram, 1936
https://www.newspapers.com/article/fort-worth-star-telegram-family-bible/143277686/