The will of Sophia Broadway appears in Davie Co. will book II:207, written 26 Mar 1897 and probated 14 Apr 1897. According to Jo White Linn's book of Davie Co. will abstracts, it names son William Thomas Broadway as executor and also mentions son Burgis Broadway "now living in Miss." I haven't seen the complete will.
Presumably this is your W. Thomas Broadway. Can you tell me more about his parents? What was Sophia's maiden name, who was his father, and where did they live before coming to Davie County? (They aren't there in the 1850-1880 census records).