Allow me to jump in here.
There is no E. L .Lawson or Dawson listed in the roster of the 6th NC that I could find.
Depending on the document/s in which he is recorded, you will find that transcriptional errors abound in old documents, especially those CW extant. Papers were written in ink, paper quality was poor, many times resulting over time with ink running &splotching. Then, papers were folded top to bottom in half, then side to side in thirds so that they could be filed in file slots. Over time, many of the corners of these papers opened up into holes.
A case in point is the finding at Vicksburg of an old list of East Tennessee soldiers which was deciphered to show my ggranduncle in the 48th Tenn. rather than in the actual 43rd. I found dozens of similar errors in he total list of 77.
As to Lawson, a D can easily be confused with L and the same with 6 & 60.
Now, look at how your man is recorded in the CW Research Database which is mainly derived from the CSRs:
Alijah L. Lawson
Residence Madison County NC; a 35 year-old Farmer.
Enlisted on 4/5/1862 at Madison County, NC as a Private.
On 4/5/1862 he mustered into "B" Co. NC 60th Infantry
He died on 5/1/1864 at Columbus, GA
(Cause of death unknown)
He was listed as:
* On rolls 11/15/1862 (place not stated)
* On rolls 2/15/1863 (place not stated)
* Detailed 3/17/1863 Columbus, GA (Detailed in gov't shoe shop)
* Absent, detailed 2/29/1864 (place not stated)
I was unable to view his records in detail, however it was noted that his detached assignment was based on his "disability."