That's a city property, not a state property, right? So any thing you're reading that says N. Carolina state parks are a no-go, wouldn't apply to the city. But I guess you're wondering if the city though, might have some rule or code that said "no detectors"? Have you tried going to their city website, to see if they have a listing of municipal codes, beach/park rules, etc.... ? I mean, *certainly* there must be posted rules *somewhere*. You know, like "dogs on leash" or "no alcahol" and "closes at sunset" and so forth. If you saw nothing there saying "no metal detecting", then presto, it's not prohibited
And as for the state parks thing you allude to: just be aware that sometimes this is deemed to apply to the land parks only, not beaches. At least that's how it is here in CA . We too have "dire-sounding" verbage in our state parks wording, that has been deemed to mean "no detecting" (at least not to detect for old things, haha, cultural heritage stuff). HOWEVER, you can detect state beaches here till you're blue in the face, and no one ever cares. There's no technical reason why the same verbage that seems to apply to the land parks, wouldn't also apply to their beaches they administer. But it's simply that way. And no one questions that, and it's simply left "good enough left alone".
Heck, didn't you loose your wedding ring on that beach last week ? Better go look for it.