I would like to just pitch a tent, be near natural water, i.e., river or lake, and be mostly away from others. Are there open areas in this park, where we can just be surrounded by trees and no people? Also how much are the fees per night for 2?

no secluded spots. You are near other campers.

This park has plenty if secluded trails to hike but the campsites are not that far away from each other. They are all surrounded by trees, with plenty of shade and convenient to clean restrooms and quarter-pay showers.

If memory serves, there are no super secluded spots. There are some that are a little further in from the road, but that's about it. As for the price, I don't remember, but there is an extra charge if you bring more than one vehicle