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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.
In October, 1986, East Branch Library opened in its own 6,400 square foot building next to the East County Annex in New Caney. The total cost of construction was $270,000, of which an LSCA grant was obtained for $50,000. In May 1987, the East Branch was renamed R.B. Tullis Library by Commissionersâ Court.