I hit a wall in my research going back as well with the information stopping with Thomas. Coming forward though, I have Captain William's son Edward born 1735 and died 1790 after marrying Winnefred Sparkman and having eight children. My line comes down from the same branch as yours --
Edward's son George Thomas Sparkman 1764 - 1816
George Thomas Sparkman's son Thomas Bryant Sparkman 1802 - 1872
Thomas Bryant Sparkman's son George W. (Squire) Sparkman 1824 - 1887
Squire's son Jeremiah Monroe Sparkman 1859 - 1928 was my great grandfather (relocated to Texas)
Jeremiah Monroe Sparkman's son Irvie Milton Sparkman 1902 - 1969 was my grandfather
Irvie Milton Sparkman's son (only child) William Jerry Sparkman 1921 - 1973 was my father