I understand he died in Burke county. However his final resting place may not be in Burke county. His wife is in Caldwell county, she died in Caldwell county. Where she is currently buried was a few years earlier Burke county.
She is buried at LittleJohn Methodist Church. This is a very old cemetery. I saw no listing for her husband. On findagrave.com a note is made that her husband was Lawrence Arney not Adam Lawrence.
I reccomend trying to obtain land records for your Arney and then trying to locate the area now. Were they living in current day Burke or Caldwell? The Burke county library would be a good place to look or contact for land records.