But the news is better for the couple Robert Cazort & his wife Charlotte Carrington. Like I said they do have a place for a daughter of the right age group for your Elizabeth.
The children of David Cozart III & Mary "Polly" Carrington, do not have a daughter Elizabeth born c1820. This couple were married in Orange Co. NC on 14 Apr 1820. We have the names of their children with their birth & death dates and who they married: Wiley Davis Cozart, Ann Carolina Cozart, Eliza Jane Cozart (born 21 Sep 1823), Green Alfred Cozart, Julia M. Cozart, Wm. Cozart, Wm. Hubbard Cozart, James Monroe Cozart, Meekins H. Cozart, Holly Ridley Cozart & Rufus David Cozart.
Sue Stewart has some pictures of this couple and their children. Really no place for your line with this family group.