Yes, flights to JFK are still bookable early June on the DL web site as operated by their partner airline. However, at over £1,200 economy for the dates I checked, the fares are unbelievably high.
Virgin's economy price are even higher and the premium ones slightly bizarre. Out 5 June. Return 9th June.
Economy return £1,335
Premium Economy £2,512
Upper Class £2,157