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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.
Village sites and burial grounds have been located in numerous places in the county; on Ripley Creek, Harts Creek, Blundell Creek, Swannano Creek, and at Dellwood Park. Titus County was, perhaps, the last home of Indians in the Northeast Texas area, living here as late as 1845.