You should use the Hawaiian website for award availability.
One thing I do if availability is tight is to search one-way awards. Once you find a few one-way saver awards, then search for the return trip as a one way and try to match them up as a round trip. If there's no availability on your dates, the website throws a warning and sends you back to the home page so the one-way search is a little more forgiving.
Another tip is to either use the HA website or google flights to find the lowest cash fares in a month or two range and then use those dates to try and find awards; there is a strong correlation between low fares and award availability.
I see lots of availability in June for JFK-HNL right now under 60K- 35K, 42,500, 47,500 generally around the 13th-22nd.
Not sure about award flights to NRT- that you may need to call HA for.