Damon Cemetery Damon, Brazoria County, Texas Cemeteries of Texas Coordiantor: Dolores I. Bishop Information provided by the State of Texas Atlas Site Location off SH 36 about 7 mi. north of West Columbia in Damon Marker: One of oldest cemeteries in county. Started Dec. 13, 1837, with burial of Sylvester Winn Damon, a relative, on Abraham Darst's land grant of 1829 from Mexico. Locating of this first grave was within a year of date of founding of Brazoria County. Cemetery dedicated and 2.5 acre tract deeded to public, Sept. 1, 1908. (1964)