In Florida, it's not legal without a permit.
Within 3 nautical miles, it's Florida coastal waters, and you would need a permit from Florida's Department of Environmental Protection as well as the U.S. Army Corps of Engineers. They will likely ask you a lot of pointed questions about why you are dredging. If it's over a reef (protected area) or right next to the beach, the answer will likely be no; if it's to salvage, the answer will also likely be no - you need a special permit to professionally salvage treasure in submerged areas, modern or otherwise.
That being said, it's a legal gray area, metal detectorists, YouTubers, etc. often go into the water to detect in non-leased areas and find modern and antique items. Technically illegal, but it is more like "unregulated" and "unenforced." The Florida artifact law exists likely so that the state can confiscate any finds they consider noteworthy after being recovered-- also to prevent wanton looting that was present in the Mel Fisher Era. Even if you are looking just for modern stuff, if you come across an artifact in submerged lands (below low-tide line) coincidentally, you are required to stop all operations and notify the state, and obtain a proper salvage license to continue. If it is an isolated find, the state will probably let you keep it; if it can be tied to an archaeological site, however, they probably won't. The beach is a mixed debris zone, so people have argued in the past that the coastal zone destroys archaeological provenance, not sure how that holds up today.
Enforcement is sporadic and infrequent, but you run a very high risk if you are dredging and doing it regularly. If you are simply diving with a metal detector in non-leased waters, no one will likely care unless you find artifacts, in which case, you would need to report them and obtain a proper license to continue in that spot. Also, nothing near national or state parks. Some of them allow certain beach and offshore activities, but certainly not inland, not without explicit permission from a park ranger. The state's interpretation of artifact is very broad (any man made object older than 50 years), but generally they only really care about Native American, Spanish, or early colonial stuff. That is my interpretation of the existing laws after a bit of research, feel free to amend or add on, if any other users with a better understanding come along.