We both seem to be researching the Independence of Texas era in relate to the Barnhill/Barnwell men who volunteered for Texas right now.
There apparently we three men with similar last names, John N. Barnhill who is said to have been from Alabama and killed at Goliad and two Barnwell men, James H. Barnwell and John H. Barnwell from Georgia who some records claim both survived.
For certain sure that James H. Barnwell survived and was discharged from the Republic of Texas army because there is a wealth of information at the General Land Office of Texas so take a look at those records.
There is a discharge record and court of claim records from his heirs with their names listed although have not had the chance to look them over yet, but yes, he was from Georgia as you have stated.
Now what is interesting is that historical accounts for both John H. Barnwell and James H. Barnwell state that after being captured at Victoria on 23 Mar 1836 they were detailed by the Mexican Army to build a boat and later escaped.
Wonder at this point if John H. Barnwell and James H. Barnwell were not the same person because you will find records for James H. Barnwell's heirs but you don't for John H. Barnwell whose heirs apparently never file claim to lands in Texas?
Which brings us to the Barnhill/Barnwell name change in different states and what you will find is that in certain records the name might be spelled either way.If you look at some of the family trees posted apparently a James Barnhill connected to William Barnhill, Sr. changed his name to Barnwell and this line continued with this spelling to the present.