Your chances are not good at all. Go as early as you can, have a back up plan! We went to the Kancamagus on a Friday in July last year and we were turned away along with other campers in front and behind us. Once you get to these campgrounds you basically have no cell service so you need to get off or almost off the Kancamagus Highway before being able to come up with a "Plan B".
When we went and all the campgrounds were full so we started calling other campgrounds and we found most of them near the Kancamagus were all full also. We ended up at Whit's End Campground - Such an appropriate name too! We were at our Whit's End when we called them!
So just be sure to have a Plan B!