I have not found any information on Elisha before he was married and had his own household. Census reports before 1850 only listed heads of households by names. There were only 2 Anderson Families in the Burke County NC area in 1800 around the time of Elisha's birth in 1794. One had a male head of household but no male children in the right age range. There was a female head of household that did list a 2 male children in the under 16 age group. Her name was Martha Anderson. I do not yet know if this was Elisha's family or not. Martha was later listed in the SC census records.
The first record I show of Elisha was 1830 as head of household. I have not gotten a copy of that census yet but am planning a trip to a large library this week that has a genealogy research center. I hope to get a copy of that record and a few others while there.
Here is a link to a familysearch.org search that I did that lists Andersons from 1800 - 1840 in Burke County NC. I'm not sure if it will work for you or not.https://www.familysearch.org/search/records/index#count=20&a...
I am still in the early stages of my research.
Maybe this will help you along the way for your research.