Individual Details

Elijah is not in all databases as a child of Jesse of Mary Wells, however his will in Wilson Co, TN definitely lists his mother as Mary Adamson.

Elijah married Susannah Hathaway, 24 Apr 1822, Lebanon, Wilson Co, TN. They are said to have had three sons, although the following will specifically says "my two sons". If the online database is correct, there were first-born twins, and probably William did not survive since he is not mentioned in the will.
William, b. 10 Mar 1824, Wilson Co TN
Jesse John, b. 10 Mar 1824, d. 17 May 1851, Fayette, IL. [He was not in the Fayette Co IL 1850 census.] Married Elizabeth Blankenship, 6 May 1847, Fayette Co, IL
Joseph, b. 9 Aug 1826, d. 19 Jun 1900. Married (1) Martha Grider, 7 Oct 1849. Married (2) Martha Mills and (3) Mary Jane Outhouse.

http://www.adamsonancestry.com/will_menus/elijah-1799-1826.html
Wills and inventories 1824-1832 FHL US/CAN Film 430842
Wilson County, Tennessee Wills Page 376.
In the name of God Amen.
I Elijah Adamson Wilson County and State of Tennessee Being in sound mind and perfect memory do make this my last will & testament in manner and form as follows: My soul to God who gave it and my [boddy] to be buried in a decent manner my wish and will is that if my just debt can be paid and of my personal property and then should be so much left. I give and bequeath unto my beloved wife Susannah Adamson a red faced cow, one feather bed & furniture it is my will that she should live on the land and log house as long as she lives or remains my widow but when married or dead the property to be sold & the money equally divided between my two sons Jesse & Joseph Adamson also the land if saved to be equally divided between them two. It is also my will and wish that the cleared land be equally divided between my mother Mary Adamson & Susannah Adamson my wife during the lifetime of my mother and do by these presents appoint Admin: Simon Adamson and Jesse Pugh executors to this my last will and testament & do revoke all other wills heretofore by me [maid] in Witnesses where of I have hereto set my hand and seal to this my last will and Testament this 11th day of October 1826.

Signed, sealed, and acknowledged Elijah x Adamson [his mark]
Simeon Hathaway [seal]
George R. Pugh [seal]

Recorded 24 Dec 1826

located in Wilson County Tennessee Will Books
ELIJAH ADAMSON Inventory. 24 December 1826. Administrators: Simon Adamson and Jesse Pugh. (Pp. 352-353) Likely Simon who married Sarah Cox and lived in Wilson Co. A Jesse Pugh married Polly Adamson, 28 Arp 1835 - she may have been Simon's daughter.
ELIJAH ADAMSON Sale. Recorded 28 May 1827. (Pp. 451-452)
SUSANNAH ADAMSON Widow's Allotment. 6 January 1827. (P. 441)
ELIJAH ADAMSON Settlement. (Pp. 271-272)

Birth	1799
Death	Dec 1826	Wilson County, Tennessee

Father	Jesse T. ADAMSON (1759 - 1825)
Mother	Mary Wells (1757 - 1825)
Sibling	Charity ADAMSON (1778 - 1828)
Sibling	John ADAMSON (1780 - 1806)
Sibling	Margaret ADAMSON (1782 - )
Sibling	Joseph ADAMSON (1785 - 1854)
Sibling	Simon ADAMSON (1786 - 1870)
Sibling	William ADAMSON (1787 - )
Sibling	Jesse ADAMSON (1788 - )
Sibling	Mary "Polly" Adamson (1790 - )
Sibling	Wells ADAMSON (1793 - 1838)
Sibling	Elizabeth Adamson (1797 - )