The second one married to Sarah C. Beard in 1870 is not him either. I have a copy of the marriage certificate and he married Sarah W. Foster in 1855. I have found her and my great grandfather Calvin, then just 13, living with Sarah's sister in 1870. William J is not with them. The census was taken in July. In November of the same year she remarries.`
There is a tax census of Claiborne county that shows a William J Beard but since I know there is a younger one in the county also, I can't tell which one it is.