Atwood Willliams was a son (probably the fifth child) of Isaac Williams & Sarah (Denison) Williams of Stonington, Connecticut. He was baptized 16 Apr 1723 in Stonington, Connecticut.
He married 3 Aug 1749 Elsebeth "Elsie" Gallup in Stonington, Connecticut. - parents of Col. Isaac Williams born 1756
There may have been additional children of this marriage as listed by Wheeler (1900) History of Stonington:
Priscilla Williams, b. 27 Jun 1761
Samuel Williams, b. 19 Mar 1764
Sources
Denison Genealogy: Ancestors and Descendants of Captain George Denison; E. Glenn Denison, Josephine M. Peck and Donald L. Jacobus; Gateway Press, Inc. Baltimore, MD 1993 [Originally published, Pequot Press 1963; Reprinted, Baltimore, MD 1978, 1982, 1993], no. 138: https://archive.org/details/denisongenealogy00deni/page/6/mode/1up
The Barbour Collection of Connecticut Town Vital Records (NEHGS, Boston, 2011) Vol. Plainfield, Page 144. m. 3 Aug 1749 Plainfield, Connecticut
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DNA Connections
It may be possible to confirm family relationships with Atwood by comparing test results with other carriers of his Y-chromosome or his mother's mitochondrial DNA.
However, there are no known yDNA or mtDNA test-takers in his direct paternal or maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Atwood: