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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.
At 53, Jack Skeen, a former Tyler city attorney, municipal judge and assistant district attorney, is nearing the end of his second decade as Smith County district attorney. He is a fifth-term incumbent who has never faced an opponent. He has a record for winning.