Callaway is a city in Bay County, Florida, United States, and is a suburb of Panama City. The population was 14,405 at the 2010 census. It is pa…Callaway is a city in Bay County, Florida, United States, and is a suburb of Panama City. The population was 14,405 at the 2010 census. It is part of the Panama City–Lynn Haven–Panama City Beach Metropolitan Statistical Area.