i dont know if its the same Cornelius but my family has somehow owned Civil War discharge papers for Cornelius Degraw for many decades. Apparrently my grandmother picked them up for my father when he was younger in a thrift store(probably in Brooklyn NY).
Cornelius Degraw a private of Captain Garret Brady Company C 2nd Regiment of New Jersey infantry volunteers who was enrolled on the 27th of May 1861to serve three years during the war is hearby discharged this 13th day of March 1864 at Brandy Station VA by reason of having re-enlisted. Cornelius stated he was born in Orange County New York, is 30 years of age 5 feet 7 3/4 inches, light complexion, gray eyee and light hair by occupation when enrolled a shiptrimmer.
Does any of this match up with your Cornelius Degraw?