Jones-Dallas, as for the 2nd part of your question (about "digging on public land"), it depends on what you mean by "digging" and it depends on what you mean by "public land".
"Digging" goes on at public beaches all the time, does it not? Kids make sand-castles, etc..... Now as for in-land terra-firma, here's where it gets tricky. Because needless to say, I suppose that on ANY public land, anywhere (schools, parks, forests, campgrounds, rivers, street-right-of-ways, etc...) that there is always prohibitions forbidding "destruction", "alterations", "defacement", "vandalism", "digging", etc...... right? And ask yourself, why is that? Well obviously because no entity wants damage to their property. Because to hone it down to the clause "digging" (although other terms like "alter" and "deface" are sometimes also used), WHAT IS THE CONNOTATION? Holes of course. In other words, the verbage distinctly and inherently refers to the END RESULT of holes. But you're not going to leave holes, right? So technically speaking, if you leave no trace of your presence (cover, stomp, fluff back up), then you haven't destructed, altered, defaced, vandalized, or dug anything, now have you? Is that to say that a busy-body lookie-lou might not gripe about the temporary interim evil process of extraction? SURE! Avoid them, and go at low traffic times.
People metal detect in public parks and school yards ALL THE TIME. A simple quick look at an "finds" bragging forums, and you'll see no shortage of people posting their latest finds, that quite often came from various public parks, campgrounds, beaches, school-yards, etc... And I bet you all of which have rules forbidding destruction, alteration, digging, etc....
Example: in my city, I can go to any park or school, presuming I'm not waltzing over people's picnic sites, or going while school is in session, or being an utter nuisance in front of park personell , and no one will pay me any mind. But sure, if I were to walk in to city hall here and say "Hi, can I dig in the park please?" or "Hi, can I alter and deface the park please?" what do you think they would say? Of COURSE they would say "no". And .... heck ... .maybe even have verbage which disallows such things. But as you can see, that's sort of like asking "can I pick my nose?". No one asks that (even though technically you might run afoul of "annoyances" clauses, or be told you need a "parade permit", etc...). No, on the contrary, they just pick discreet times, and don't make a big scene.