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Sweeny, Texas. Sweeny is a city in Brazoria County, Texas, United States, the westernmost incorporated town in the county. The population was 3,684 at the 2010 census. The city's motto is A City With Pride. The city is also known as Adamston.
City of Sweeny, Texas | A City With Pride | Welcome to our Official Web Site. The City of Sweeny is proud to present this Web site for our residents and the surrounding community. Our motto is A City With Pride.. We are proud of our history, our government, and our community.