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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.
Sweeny, TX Profile: Facts, Map & Data. Sweeny, Texas - Basic Facts. The City of Sweeny had a population of 3,920 as of July 1, 2016. Sweeny ranks in the upper quartile for Population Density when compared to the other cities, towns and Census Designated Places (CDPs) in Texas. See peer rankings below.