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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.
The establishment of Sutton County set off a new round of competition between the interests at Winkler's Well, now called Wentworth, and Sonora, over the location of the county seat. Initially Wentworth had the advantage, but in 1889 Charles Adams was successful in drilling a well on the Sonora courthouse property.utton County (J-11, J-12, K-11, K-12) is in west central Texas.