There are 1 US Post Offices in Sutton County serving a total of 4,180 residents equating to 4,180 residents per post office. It's estimated that approximately 6,672 packages are handled across the 1 county post offices annually. Sutton County is a county located on the Edwards Plateau in the U.S. state of Texas. As of the 2010 census, the population was 4,128. Its county seat is Sonora. The county was created in 1887 and organized in 1890. Sutton County is named for John S. Sutton, an officer in the Confederate Army.