I'm with mayberry-monte on this one. Break down your question into 2 parts:
a) Johnny, you say you "
...called the local school board..." You're talking about a
public school(s) in this part? If so, then permission not needed. If the school is open for others afterschool hours (joggers, monkey-bars, basketball, etc...) then think of yourself as no different than any of them. If you intend to leave no trace of your having been there (everything re-packed, stomped, fluffed, etc...) then the "digging" thing is a mute point IMHO. If you're there after 5pm, or on weekends, I bet no one cares or would've noticed you (unless you were being a nuisance, sticking out, leaving holes, etc...)
b) as for "....
a private academy and a private Catholic school..." Yes they are technically private. But so too is a shopping center, for example, & no one is prohibited from waltzing through. Ie.:
open, but "Permission to pass is revokable by owner..." I went to a parochial school up till 6th grade. After school, we would return to the school yard ... as it was sort of like the neighborhood playground. Then in 7th grade, when I got my first metal detector (although I was now in public Jr. High) the natural place to go try it out, was at this nearby parochial school I had attended years earlier. It never even occured to us (although the mid 1970s may have been a different climate?) that anything was wrong. It was open school grounds, and seemed open to anyone else after hours. No one ever said anything to us, and we mined lots of silver coins

Now I suppose if a nun had come out to tell us "scram", then of course they could've. But no one ever paid us any mind.
But the scenario in b-above has many many other factors where common sense comes into play. Like: is it fenced or posted? Are others using it afterhours (basketball, jog, etc...). Are we talking historical monuments? Super-groomed turf @ an uppidty up institution? or small-town back-lot scruffy school sandbox? I mean, it's no different than decisions we md'rs make for a public site too: sometimes you just have to use common sense, to know when you need to go through bureaucracy, verses when it's obvious no one could care less. Ie.: some public parks you can just "feel" that you'd be a pink elephant walking around, so you skip them. Others you can tell they're low key.