From The Beckwiths : "...she was the daughter of Moses Seymore of Litchfield; and a niece of Governor Haratio Seymore of N.Y." (Notice the different spelling of the last name.) Mother was Mabel Strong.
Children:
Jane Seymour - died at 4 year old
Elizabeth Gale - never married
Josiah Gale II - also a doctor - m. Anne Bonfield Allsopp
George Seymour - m. Isabel Cohen
Sarah Hunt - never married
Sources
Beaumont/Jacobs Family papers, supplemented with information found online.
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DNA Connections
It may be possible to confirm family relationships with Jane 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 Jane: