Individual Details

This connection is a theory at this point - proved only by DNA suggestion. If a daughter of John, she would have likely been born out of wedlock to teenagers. There doesn't seem to be any other place for her, however, and Rachel is a very common name in the Adamson family. It is possible her sons were in error regarding her brother and father, but she does seem to belong somewhere in this family.

Rachel is plainly in the 1850 Census, Van Buren, Monroe Co, IN, age 66, b. NC, with children Labin, 25 and Martha, 20, both born in IN.
War of 1812 Pension card files has the following:
No claim for pension. Soldier: Solomon Green. Widow: Rachel. Alleged Pvt under Capt Robinson, Tennessee & Ohio Military. Bounty Land #144372-50. Fold3 has this application, she applied 14 Apr 1853, but there is no evidence any land grant was forthcoming. It does have Rachel's date of birth and the death date of Solomon. They were married 9 Jun 1804 in Jefferson Co, TN and she was Rachel Adamson prior to the marriage. She stated that they moved to Ohio after he served in Tennessee and he served again from that state and was appointed Lieutenant in the Ohio Militia. She claimed they lived in Monroe Co IN for upward of 30 years. She claimed that there was a family Bible in possession of son Labon Greene. Solomon Green was found as serving under Capt. Robinson from 28 Jul 1813 to 12 Aug 1813 - which would not have fulfilled the six-months service requirement.

Will of Solomon Green was presented in Monroe Co Court, August Term by Thomas Fullerton, one of the witnesses. Sworn to 21 Aug 1849.
28 June 1846. Provisions of the will included: $5 dollars to son James, daughter Elizabeth, son Jesse, daughter Ruthy. To son Solomon he gave the West half of the NW quarter of Secion 19, Twp 8, Range 1 West. Half of his grist Mill with give acres of land to include the Mill Pond. To son Morris, $100. Daughter Rachel, also $5. To wife Rachel, parcel of land for her lifetime or widowhood - NE quarter of S25 Twp 8, Range 2 West, with the exception of the five acres bequeathed to Solomon. Also 30 acres lying in Section 24, Twp 8, Range 2 West. To son Laban, after the death of his mother (the above lands alotted to wife Rachel). to son Isacc, $100. Daughter Martha, $5. Wife to have household contents during or life or widowhood - to be divided share and share alike between my two daughter, Rught an Martha, and Rebecca Whaley. Executors - sons Solomon and Labon. Signed: Solomon (X) Greene. wit: Thomas Fullerton, William R. May
Admitted to Probate, 21 Aug 1849.

Steven Chambers (Ancestry)
Rachel Adamson is the daughter John Adamson (1767–1828) and the sister of Enos Adamson (1793–1846). 
 It took me several years since my first research to make this discovery. It began by finding or rediscovering a document listed as: 
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Transcript
We Solomon Green and Laban Green Executors of the Last Will and Testament of Solomon green deceased report to the Court of Common Pleas the following account: .....The exccutors are chargeable with a residue of a claim against the estate of Enos Adamson deceased of the distribution share of decedents of estaate of John adamson deceased in right of decedents wife formerly Rachel Adamson & daughter of said John Adamson............... ------------------------------------------------- clearly states that Solomon Jr. and Laban Green, sons of Rachel Adamson/Green believe and hold that Rachel is the daughter of John Adamson and implies that she is the sister of Enos Adamson.
Detail
No. 40 No. 27 Filed January 20th 1853 Milton McPhetridge, Clerk
Source information
Title
Probate Record Monroe County, Indiana Court
Repository information
 
After this finding I still do not know which John Adamson. I used the DNA match list in Ancestry to search Adamson’s last names. I found several that all matched this Adamson line and luckily those DNA proven cousins had shared not only their DNA but done their tree’s properly and went far back enough in time to compare them. I am 100% convinced this is the correct line. 
John Adamson (1767–1828) would have been about 16 at the time of Rachel’s birth. She is not listed in his children’s records. She would have in fact been his 1st born child, possibly out of wedlock and more research is needed now that this line is proven. DNA will help further the research through comparisons to other living individuals that share this line. 
The DNA and record lines I have found so far have no other visible connections at the time of this writing. 15 Oct 2021. 

My thoughts: The Newberry records say that John Adamson died at the home of his son Seth on 12 Jan 1828. He married Mary Hammer in 1786 - three years after the birth of Rachel. They did indeed have a son Enos - a name found in several Adamson families.

Birth	23 Nov 1783	North Carolina
Marriage	9 Jun 1804	Jefferson County, Tennessee - Solomon Greene
Death	Aft 1853

Spouse	Solomon Greene (1772 - 1849)
Child	Isaac Greene (1820 - 1870)

Father

John N. ADAMSON (1767 - 1828)