I don't have anything more about Tilman S. White's parents or death dates, but on the 1870 census he was a farm laborer in the household of James Williams and Mariah Peebles in Polk County on page 24, family 171. Their daughter Georgia Ann is in the household listed as Georgiana.
His birth date and place are different on the 1860, 1870 and 1880 census. In 1860, he's 27, born in Georgia. In 1870, he's 40, born in Georgia. In 1880, he's 40, born in Alabama. The closest to a match I've found for any of these listings on the 1850 census is a Tilmon D. O. White, 16, born in Georgia who was residing in Jackson County, Georgia.