Southern, fares in the US seem rather haphazard, and depend on specific day and time of travel, and how early you book. I would look in or to get the best prices on the exact date you plan to fly. You may be surprised to find that the lowest fares may not be on the "discount carriers" but on the majors.
Also, as noted, you may try the other NY area airports, if you have ground transportation available. But beware of heavy road traffic and possible delays, not to mention the cost.
Another caviat is that if you buy a discount ticket separate from your overseas ticket, plan to connect straight through but miss your flight to Washington due to a late arrival into JFK, you may have to pay a fee for reissuing your Washington ticket. If, however, your overseas ticket includes the segment to Washington, then there would not be such a fee. Smilarly, you can get a "refundable" fare to Washington, but that will be more expensive.
"In God we trust, everyone else bring data"