Even though DL is noted for frequent schedule/aircraft changes, I think I would get it sorted out sooner, while the chances of seat availability (not award seats, just F seats) in F are greater. DL has 2-3 Flights Daily PHL-JFK, so, depending on your arrival time, you may have options on DL:
Code:
DL5042~ PHL 1005 JFK 1100 every day
DL5548~ PHL 1435 JFK 1535 Sat.
DL5548~ PHL 1545 JFK 1700 Sun. thru Fri.
DL6098~ PHL 1840 JFK 2005 Sun. thru Fri.
I have AF366 which arrives in PHL at 15:35, and was originally scheduled to depart on DL6098 departing at 18:40. It seems like the flight was cancelled, and they scheduled another one DL5060 which departs PHL at 16:00. It seems that is the last PHL-JFK flight DL has that day (July 14).
Edit: It certainly wouldn't hurt to call the PE line now, if there are further changes you can call again.
BTW, there is a DL CRC in PHL, which is now inside security. AF comes into A West and DL and CRC are in A East, and terminals A through D are all connected inside security.
I did notice that when I looked at the location of each airline on the PHL website. Even assuming optimal conditions, I doubt that it would be possible to get my bags, and clear customs/immigration in time to make the DL flight.