I've just researched a Pergerson line for a friend.George I Pergerson of Franklin Co NC, 1900, b 1886, son of James Richard Pergerson, b 1844, on the 1880 Franklin Co census, son of Joseph Pergerson b 1800, Granville Co NC census, son of John Pergerson? 26-44 in 1800, who may have been the son of Joel Pergerson, over 45 in 1800, both Wake Co 1800.
In 1830 I noticed a Joel Pergason in Halifax Co Va, b 1770 to 1780. He may be the father of your Halifax ancestor.
I did find Perkinson in Warren Co NC, 1810 and 1850. they were indexed Pergason in 1830 but are clearly Perkinson in 1800 and 1850.The Perkinson you found in Charlotte Co Va is a likely ancestor.
Not my family, just a random act of genealogical kindness.