Since the Welsh used no family names until the late 1600's and early 1700's, people who lived before that time had no family name. It is probable that your ancestor was the son of Thomas, the son of Cadwalader - in Welsh terms: Thomas ap Cadwalader. When the Welsh left Wales for the new world they assumed which ever second name they carried at that moment as a family name. The first Cadwalader in our family to arrive here was John ap Cadwalader ap Hugh ap Thomas, etc.
When he landed in Philadelphia he used it as John Cadwalader. His uncle, Thomas ap John, became the head of the now very numerous Jones Families of Central Pennsylvania.
Try to let me know when your ancestor arrived here, I may be able to help.
Robert N. Cadwaladercdwldr@aol.com