In the book, Marriages of Jefferson County, Tennessee, 1792-1836, compiled by Edythe Rucker Whitley, I found Elizabeth Neal married Thomas Elmore Dec. 22, 1822. William Cox married C. Austen June 30, 1814. These people are of the right time period for your William and Elizabeth, so I thought I would let you know what I found. They might be the ones you are looking for, although they weren't marrying each other at this time. I hope this helps.