I am curious why you think John F Womax's grandfather was named William?
John F Womax's parents were Jesse Womax/Womack and Lucinda Miller. Besides being in Shenandoah Co, VA in 1850, Jesse "Wamux" was on the 1820 Census of Fauquier Co, VA. I cannot find Jesse in 1830 or 1840. He married Elizabeth Malin on 6 May 1818 in Shenandoah (his name given as "Jesse Wamix"). He married Lucy Miller on 13 Aug 1844 in Warren Co, VA (as Jesse Wommack).
I am pretty sure Jesse was the son of Peter Womack and Christina Utterback, who married 29 Nov 1786 in Culpeper Co, VA. Peter Womack had a brother named Jesse. Peter appears on tax lists of Culpeper and Fauquier Cos, VA in the late 1700s and early 1800s.
Let me know if this fits with your info, and I will continue with Peter's ancestry.