Clint, If I am stating the obvious, then I apologize...but here goes.
1. Put a post out on the Twiggs County forum and see if Twiggs County has published any deed books, wills, marriage records, etc.Fathers often deeded land to their sons once their sons reached a certain age.
2. Posters on this forum have mentioned a couple of Asbills that moved to Twiggs Co. from SC.One of these Asbills was Elisha,whose father actually moved to SC from Bertie County, NC.I have found a listing that the father, Aaron, was born in Perguimans, NC. Aaron had another son, John, who also moved to Twiggs from SC. All these guys and your Jesse seemed to have hit Twiggs about the same time..so my guess is that they were cousins.One of the posters may have run into your Jesse, so respond to their post and they will get a "hit" in their email.
Jesse naming his son, James, may lead to the father or an uncle...The Asbill men did (and in my immediate family still do) like naming their sons after Asbill relatives.In SC at least, the middle name is also very important.My Dad's middle name is the same as his grandfathers middle name, which I have traced to a marriage in 1810.