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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 East Branch Library was opened in 1974 in leased space in a shopping center between New Caney and Porter. In March, 1983, the East Branch Library was moved into the East County Courthouse Annex.