Ron, I notice from this last note of yours that Gainesborough is spelled both with an "e" and without an "e" - any idea from the context of the letter or other occurances of the name which one is correct?
I am beginning to think that Gainesborough or Gainsborough may refer to a Mr. Gainesborough's farm or country store or other some such thing instead of a community or town?
If so, it would be located as mentioned before (from the OR) on a road to Carthage in or very close to Leake County.