Possible marriage to Margaret before Sarah Hawkins. Found Lucinda but none of the other children listed. Added the census record below to use for further research.
Sources
"United States Census, 1850," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MCDW-2YM : 12 April 2016), William Cook, Bradley county, Bradley, Tennessee, United States; citing family 1125, NARA microfilm publication M432 (Washington, D.C.: National Archives and Records Administration, n.d.).
ID: I606714358
Name: William COOK
Given Name: William
Surname: Cook
Sex: M
Change Date: 6 Oct 2003
Marriage 1 Sarah E. HAWKINS b: Abt 1792
Children
Has Children Lucinda COOK b: Dec 1829 in Montgomery,AL
Has No Children William R. COOK
Has No Children Arthur R. COOK
Has No Children Green COOK
Has No Children Ann COOK
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DNA Connections
It may be possible to confirm family relationships with William 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 William: