There are 2 US Post Offices in Clinton County serving a total of 10,299 residents equating to 5,150 residents per post office. It's estimated that approximately 16,438 packages are handled across the 2 county post offices annually. Clinton County is a county located in the U.S. state of Kentucky. As of the 2010 census, the population was 10,272. Its county seat is Albany. The county was formed in 1835 and named for DeWitt Clinton, the seventh Governor of New York. It is a prohibition or dry county."