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The closest equivalent to the British Strict Baptists in the US is Primitive Baptists, also called 'Old School Baptists' or 'Hardshell Baptists'. Like the Strict Baptists they practice believer's baptism and closed communion, take the Bible as their only doctrine, and eschew seminary training (they also oppose non-local missions). This theological statement by an Atlanta Primitive Baptists congregation gives more detail.
Related groups include the Missionary Baptists. I've been unable to discover if they have closed communion.
The List of Primitive Baptist churches linked from the Wikipedia page gives a list of around sixty congregations throughout the US. Since the Primitive Baptist movement consists of independent congregations with no control or accountability you may find the congregations very different. You should probably be warned that one of the congregations listed has a bad reputation nationwide for extremist inflammatory hate speech.