Ann was born about 1745 and passed away about 1805.
I have found no record of Anna (Hannah) Avery, the wife of Josiah Avery. According to a DAR record, she was born in 1745 and died in 1805.
Several unsourced family trees provide information about Anna. See, for example, Chadwick-1021 18:13, 8 January 2017 (EST). 
Since "Hannah" was often interchanged with "Ann" I've added a potential reference for Josiah Avery's wife. Her name may have been "Hannah Pritchet".Fetterly-28 23:51, 21 November 2020 (UTC)Fetterly-28.
Marriage (most likely): Ancestry.com. New Hampshire, U.S., Marriage Records Index, 1637-1947 [database on-line]. Provo, UT, USA: Ancestry.com Operations, Inc., 2011. Original data: "New Hampshire Marriage Records 1637–1947." Index. FamilySearch, Salt Lake City, Utah, 2011. “New Hampshire Statewide Marriage Records 1637–1947,” database, FamilySearch, 2009. New Hampshire Bureau of Vital Records. “Marriage Records.” New Hampshire Bureau of Vital Records and Health Statistics, Concord. Citing marriage of Josiah Avery to Hannah Critchet/Pritchet on 23 May 1793, Sanbornton, New Hampshire. FHL Film Number: 1001121.
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It may be possible to confirm family relationships with Anna by comparing test results with other carriers of her mitochondrial DNA.
However, there are no known mtDNA test-takers in her direct maternal line.
It is likely that these autosomal DNA test-takers will share some percentage of DNA with Anna: