I wish I had answers to your questions about James Kenner, son of Winder Kenner. I have not found Martha's maiden name either nor do I know for sure what led him to move to Louisiana then Texas. James' sister, Elizabeth Howard Kenner McCraw and her husband left Hawkins Co, TN around the same time and moved to Greene Co, MO most likely because new land was opening up and land was becoming less available in Hawkins County.
I do have names and approximate or actual birthdates of all 13 of James and Martha's children and spouse information on 9 of them. I wonder if your Martha Thompson is related to Jane Thompson who married James' brother Francis "Frank" Kenner.
James had to have moved to Louisiana sometime in 1830 as he is listed on the 1830 Hawkins Co, TN census and yet son James was born in LA 10 Aug 1830. Their last child born in LA was Marie Minerva in 1847 so sometime between 1847 and the birth of Nancy in 1850 in San Augustine Co, TX is when they made that move.
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Audrae Turner Mathis