We have the same problem at our state parks and I imagine it is common everywhere during major holiday weekends. About half of the sites are not available for reservations at our state parks and in the past we have come a day or two early to get one of the first come first serve spots.
Whether this will work at this particular park is hard to say, but if you are deadest about camping there without reservations during this specific time period, getting there early as possible is really your only option. Here's hoping it works out for you.