I'm afraid I'm not going to be much help in your quest.
John Douglas Carroll (he had a grandson with a middle name of Dart, which may be where the IGI submitter went wrong) was born 20 Feb 1808 in Hancock County, Georgia, according to his tombstone.He was the son of Richmond Carroll and Sarah Driver.
John Douglas Carroll married first Rachael Rennick in 1828, second Rachel McGill in 1831, third Julia B. Stedman in 1837, and fourth Fernella in 1853.
He is not the John D. Carroll in Georgia in 1850.In 1850, John Douglas Carroll is listed in Carroll County, Mississippi.Land deeds and other documents show that he is definitely in the area before and after the census.
So, having said all that, I suspect the two John D.'s are related.Richmond Carroll (John's father) was the son of Douglas Carroll, who received land in that area of Georgia.He had four sons:Alfred Allison, Harvell Hardy, David, and Richmond.I haven't done much research on the sons besides Richmond.The John D. Carroll in Georgia in 1850 may be the son of one of them.
But none of that answers your question of whether Sarah Carroll is the daughter of John D. Carroll listed in 1850 Hancock County, Georgia.I don't have anything on that.Sorry.