I assume that Frances Sanders is the "Fanney Sanders" found on the 1850 census for Lawrence Co., Myatt Twp. AR. Is the John A. Sanders found on the next page (Strawberry Twp.) her brother-in-law? He too had a son named Major H. Perhaps this is why Frances moved her family from Searcy to Lawrence after her husband Major died?
According to the stated birthplaces of his children, John moved his family from TN to AR between 1834-36. His household in 1850 is listed:
John A. -- 45 (1805) TN. Keziah -- 51 SC. Major H. -- 16 (1834) TN. Mary -- 14 (1836) AR. William -- 11, AR. John P. Sanders -- 11 (or 21) TN. Charlotte -- 10 AR.
The entry for John P. Sanders is interesting. That his full name is given seems to indicate he was not John's son, but possibly a nephew. Ancestry.com indexed his age as 21, but the actual text looks more 11. If he was John's son he was a twin of William. I think it more likely that he was a nephew. And because Charlotte was listed after him may indicate that she too was not John's child, but a niece. Who's children were they?
It is noteworthy that John moved his family from Bedford to AR during the same timeframe that the family of my George W. moved from TN to AR, alhtough they went to different areas of the state.
Your list of children attributed to Major and Frances does not include a son named Joseph. The 1850 census shows someone in her household named Joseph, age 20, b. TN. Again, because he is listed last my indicate that he is not her son, but possibly a nephew. This makes me wonder if he is a brother to the John P. and Charlotte living in the household of John and Keziah.
During the Civil War a company of soldiers was raised in Lawrence and Sharp counties in 1862, and became Company D of the 38th Arkansas Infantry Regiment. The captain of this company was William J. Sanders, age 35 (abt. 1827) born in TN. This appears to be the son of Major & Frances. 3rd Lt. was Major H. Sanders (son of John & Keziah), age 29 (1833) born in TN. William M. Sanders was a sergeant in this company. No age was given but he was born in AR. He also appears to be a son of John. Lastly, there was a Joseph Sanders, Pvt., for whom no further information was offered. This appears to be the Joseph found in the household of Frances in 1850. The fact that William J. was elected captain (commander of the company) and Major and William M. were elected officers, indicates they were prominent in their community, and probably all related.
In your reply you stated that Major moved to Bedford after 1827 to an area just south of Shelbyville. Shadrack and Tabitha Hall Brown, the parents of Mary Magdalene Brown, mother of my great-grandfather George, lived in this exact same area.
I'd liked to keep digging on this but am pressed for time at the moment. Perhaps it would be easier to continue communicating by email. Please feel free to contact me at firstname.lastname@example.org