The majority of the Shaws in colonial Cumberland County , New Jersey were descendents of EDMUND SHAW
Son of Richard Shaw and Remember Shaw Husband of Rachell Shaw Father of Edmund Shaw; Richard Shaw; Joshua Shaw; Caril Shaw; Mary Shaw and 3 others Brother of Richard Shaw; Joshua Shaw; William Shaw; Benjamin Shaw and Elizabeth Shaw
several histories of New Jersey founders are available to read free online
http://www.westfieldnjhistory.com/files/lrgenes.htm
Although many been looking, no one has been able to find a record for the parents of Cornelius Shaw.
I've summed up all the sources I can find on his profile here: http://www.wikitree.com/wiki/Shaw-6508