I have looked for John Garner's obit, but so far no luck.
I've found no link between John Francis Garner & James Knox Polk Garner. James' parents were William & Sarah Garner, from Lewis County, TN.
They came to Texas with their slaves, who took their surname. Mitt Garner is descended from those slaves.
James married Nancy Scott Whiteley, and they had 12 children:
1.Samuel Jacob Garner m. Cordelia Parker, m. Addie Courtney
2. Sarah Tennessee "Tennie" Garner m. Bob Newsome, m. James Jones
3. Harriet Ellen Garner m. J.E. Jones
4. Frank Wilson Garner m. Daisy Heartfield
5. Martha Garner m. Ennis Moore
6. Mary Garner (twin of Martha) m. Virgil Herring
7. George Washington Garner m. Euda E Stewart
8. Emma Lou Garner m. Harvey A Crister
9. Willie Homer Garner m. Exie Lee Stokes
10. Robert Luther Garner m. Winnie Smith
11. Ella Dewitt Garner m. Tom Dee Leon Stokes
12. Claude Roger Garner m. Willie Janet Vestal
The above info is from Story of Bell County, Texas, Vol. II