My approach would be to set up a plan of specific dates and start checking often online now. People do cancel, and maybe you can get a night or longer. Sometimes once you are "in" and are physically there, the rangers on site are forthcoming about inventory they control close in.
Then scout the back-up possibilities, fanning out for 10-30 miles or so, for commercial campgrounds, state parks, Army COE, etc. Some take res, some are first-come.
Do you have an app like AllStays? It's very handy for finding things in a specific vicinity.
I've found that late August is still prime time. Families are thinning out for back-to-school, but there are still many Europeans in rentals, and of course others without kids.
