Summer flights are notoriously high from EWR and JFK, so many Dominicans go there after school closes, the flights are packed going in July and coming in August.
When I worked in the NY textile industry I had a mandatory 2 week vacation for the first 2 weeks in July [whole industry closes down - so did the whole NY jewelry industry where my dad worked for 40 years], and I booked flights almost a year in advance to get a decent price.
You might want to call some of the charter/tour companies that service the NY area, like Apple Tours, Liberty, etc. for example, and ask if they will sell Air Only to Punta Cana. There are a few tour operators, and they all have blocks of seats on flights that they have to fill. You'd have to fly on their tour dates, usually weekends, but it might be worth a shot.