The first things I would do, if you haven't done these already, is to :
1. Get a copy of JH's marriage record. Some old records list the parents of both the groom and bride.
1. Get a copy of his death certificate to see if his parents were listed there.
3. Search the Clerk of Court's office for both Claiborne AND Union Parishes for any guardianship, tutorship, probate, and land records for James. I have found the guardianship records in Claiborne Parish are a goldmine of information when a parent dies and leaves minor children.
I would look closely at the J. L. Williams family in Claiborne , 1870, even though the age is off a bit. There are two 18 yr olds and two 16 yr olds but not listed together like twins would be. James is one of the 16 yr olds. I think this is an indication that some of these children are not theirs. Problem is that there's no sister that matches your information. Maybe she was older and already married.
Good Luck in your research.