I think that private campgrounds and RV Parks are going to be more limiting and less flexible on this. Some, like WRVPO, will simply make a rule and enforce it to eliminate arguments and "exemption creep". Others will see it simply as a way to make more money.
Public campgrounds quite often have a "family" exemption. That is usually defined as 1 or 2 parents and their immediate children of various ages. It usually not mean grandparents, adult kids, and their individual families.
The concept is that most public campgrounds don't want a bunch of adults in one site (usually ends up as a "party") and they don't want kids under 18 unsupervised in a site (potential for accidents, etc.). So, if you are a true family (which the OP is), the campground will waive the person limit.
If the kids are all boys or girls and wearing funny uniforms, let they probably won't accept you as a family.
I would simply call before making reservations and ask if you would have to pay the extra fee. I suspect most public campgrounds will waive it. If they don't waive it, then you can choose to pay the extra fee or not. If you need a site last minute, then you may not have a choice for that moment.