BA only guarantees releasing 4 Y and 2 J award seats on all flights, however you should be able to find good F award availability to NYC. BA may release additional award seats close in to departure.
All you can do is keep checking for availability. You could always book CW and then change if F award seats come up.
Very little, to the point where I'm actually wondering if I'm searching correctly. According to (
https://www.britishairways.com/travel/redeem/execclub), there are maybe 10 flights that month with any availability, and only one, at the start of june, with 2 seats.