William Swanson is named in the Will of his father, Joseph Swanson, Sept 1833, whereby his son, Otho has qualified and is named as Executor of the estate. (see below)
Along with Joseph's wife, Susannah P.(?), he names the following children: Otho, Wm., Jno T.(?), Jas.M., Chas S., Eliza J., Louisa, Joseph, Francis, and George W. Swanson.
Also, I found reference to Wm. Swanson with regard to a Will dated 19 Mar 1834 for a Othello (Otho) Swanson. He bequeathed his land given to him by his father to be sold and any money to be used to purchase land more convenient for his mother, Susannah Swanson and his brothers and sisters. He does not name them. However, in Sept. of the following year: Administrators de bonis non bond, Susannah Swanson, Wash Hill and William Swanson for Susnnah Swanson.
Hope this lends some clues for you.
Will Book, Amherst Co., Va 1761-1865 (look-up)