There are 7 US Post Offices in Claiborne County serving a total of 34,062 residents equating to 4,866 residents per post office. It's estimated that approximately 54,367 packages are handled across the 7 county post offices annually. Claiborne County is a county located in the U.S. state of Tennessee. As of the 2010 census, the population was 32,213. Its county seat is Tazewell."