Many of the names in your post reflect names in my Beachum family of Anson County, NC, as well as the location Henderson County, Tennessee.
Family legend has it that the name Beachum was originally spelled Beauchamp. In the land and court records, the name was variously spelled Beacham, Beachum, Beecham, Beachim, etc.
My fourth great grand father was Alexander Beachum (whose name was also written as Alex, Alec, Ellick, etc.) The first record I have found of him was of his selling land in Anson County, NC in 1796. There appears to be no record of his having obtained the land. (The Anson County Court House burned in 1869, and many records were lost.)
The 1790 census of Anson County does not list Alexander Beachum, but it lists four Beachum households. Two are headed by men named William Beachum, one headed by a man named Jesse Beachum, and the fourth is headed by a lady named Susannah Beachum. I presume that Alexander belonged to one of these households.
A deed from 1814 lists Alexander Beachum and his wife Elizabeth as heirs of Beverley Shelton. I presume that Elizabeth was the daughter of Mr. Shelton and the mother of the children of Alexander who were born between 1806 and 1812, but I do not know this for a fact. I have been told that both Alexander and Elizabeth were born in Virginia.
I know of three sons of Alexander Beachum: Hugh A. Beachum (1806-1861), Daniel S. Beachum (1810-1886) and Isaiah Beachum (1812-1894). Family legend has it that some of the sons, and possibly the father Alexander, went west to Tennessee. I know that Daniel did live in Tennessee, Alabama & Mississippi. He seems to have spelled his name "Beacham."
Both Hugh Beachum and Isaiah Beachum remained in Anson County, North Carolina. They married Zillie O'Neal (1812-1882) and Phereba O'Neal (1818-1888), daughters of Kindred O'Neal (1785-1855). The family story states that three Beachum brothers married O'Neal sisters, but I have been unable to verify this.
Hugh and Zillie had the following children: Jeremiah Washington Beachum (1830-1910), Joseph J. Beachum (1833-1863), Harriett Beachum (1834-1870s), James Thomas Beachum (1837-1907), William (c. 1840), Daniel (c. 1847). There was also a daughter named Christina Clementine, who died young.
Jeremiah Washington Beachum married Mary Jane Taylor (1836-1882) in 1853 and they had 14 children. I am descended from their daughter Mary Jane Beachum, Mrs. D. B. Stearns (1867-1910).
As you can see from the above, the names Alexander and Hugh figure in my Beachum family. Jeremiah Beachum also had a daughter named Louisa and a son named Zachary, though I was always told that Zachary was named for his wife's kinsman, Zachary Taylor. It seems likely to me there is some connection.