He couldn't have been born in Wake County because it was created 1771 from Cumberland, Orange and Johnston Counties. He must have been born in one of those counties.
North Carolina didn't issue birth records that early in time.
I'd look for a copy of his marriage record to see if his parents are on it. If not, then I'd look in the local newspaper for their marriage announcement.
i'd also look for the newspaper announcements of their children. Their grandparents might be listed as well.
I'd contact the Wake County Genealogical Society, https://wakecogen.org/cpage.php?pt=32
and ask if they have any records 1757-1799.
Did you look in FamilySearch, Catalog? That's where you'll find digitized film that hasn't been indexed yet. You can still browse through the records.
I'd also contact the Wake County Historical Society, www.wakehistory.com/
and ask if they have a local death index, indexed newspaper death notices and inventoried cemeteries. His parents names might be in his newspaper death notice. Successful Searching!