I have a student named Betty Dortch in my list of Mordecai students. She's only at the school one session (first half of 1816), and the adult attached to her account might be Major John R. Eaton. It looks like a "Maj. J. R. Eaton" paid for "Miss Dortch" in the school ledger, January 1816; there's another mention of the two names together in the May 1817 ledger page, along with the name "R. Bullock." Other than that and the school rosters, she doesn't seem to have been mentioned by the Mordecais. Not much to go on, but it's an unusual enough name, let's have a look around.
It's definitely the name of a prominent North Carolina family in the nineteenth century. William T. Dortch (1824-1889) was a North Carolina legislator, born in Nash County near Rocky Mount; he once owned the historic house named the "Dortch-Weil-Bizzell House" in Goldsboro (for sale, and it's accessible!). Another North Carolina Dortch moved to Tennessee, and his son (another William Dortch) moved to Arkansas, where he owned Marlsgate Plantation. Anyway, plenty of Dortches.
Let's try the Major Eaton angle. John Rust Eaton (1772-1830) was a planter, horse breeder, and state legislator from Granville County, NC. He mentioned a "Mr. Dortch" recovering from smallpox in a 1794 letter, but that's it for the name's appearance in his published correspondence. In 1816 he would have been a father to some of his eleven kids (he married Susan Somerville in 1801). And his sister Betty Eaton married.... Noah Dortch. Bingo. Another of Eaton's sisters, Mary, married William Baskerville--we've already run into him, because Eaton had Baskerville nieces at the Mordecai school as well.
So here's the story. Elizabeth Anne "Betty" Dortch was born in January 1803, first child of Elizabeth "Betty" Eaton (1787-1810) and Noah Dortch (1781-1811). As the dates show, Betty (and four younger siblings) lost both mother and father by 1811--which might explain why uncle John Rust Eaton was taking care of her school tuition five years later. At 21, she married James Bullock Jr. (1798-1880), whose sister Catherine Bullock also attended the Mordecai School. (His sister Fanny Bullock married Macon Green, who was one of the Mordecai's male students in their early years.) Betty and James had five children together; two died in infancy. Betty Dortch was destined to follow her parents to an early grave--she died in 1832, age 29.