Was the Conkwright name ever Courtwright or courtright? I say that because I have a Courtright ancestor whose name was VanKorkryk from Belgium/Netherlands (there's a town called Courtrai or Korkryk depending on the language). Van just means "from". VanKorkryk would mean from Korkryk. It turns out that this family was originally named Cortes (and mixed with another family named Bastien/Sebastien y Castillo). They were from Northern Spain, became Calvinists and were chased out of Spain and went to the Netherlands. It turns out that there is some evidence that they were Sephardic Jews before conversion to Protestantism.
Don't know if that helps....might be a different family, but family names change a lot in America especially with the spellings going from Dutch to English.
Fr. Philip Kontos