I don't know of John Marshall having a son William. From the 1860 to 1880 censuses, his children were
John W Rigsbee 1859-12-xx Chatham Co NC Nancy E Rigsbee c1868 Chatham Co NC A R Rigsbee (m) c1877 Chatham Co NC D J Rigsbee (f) c1878 Chatham Co NC Martha Rigsbee c1879 Chatham Co NC
John and Nancy match up with your list. "D J" must be Jane. Perhaps William, Bunnie, and Bessie were born after 1880. I'm currently searching the 1900 census for NC, but haven't seen John Marshall yet. He lived until 1910, so he should be there.
At any rate, there's no way John Marshall could have a granddaughter born in 1870. His daughter Martha is probably too young to be your Martha, too.
There were not very many Rigsby families in LA up to 1900, and most of them were apparently descended from slaves owned by a Rigsby family in MS. There was also John W Rigsby, who was one of my cousins. He came to Sabine Parish from TN by way of MO before 1870. The only other I know of was a William Rigsby born in 1860 who was in Morehouse Parish in 1900.
I'm not sure how your Martha fits in, but I'll keep looking for her.