not sure about NJ.
in Florida, rules may vary by county or city but most beaches have exclusion zones that apply to any boat propelled by "machinery" (that includes electric motors). a few years ago, on calm days, I used to anchor off Crandon beach (Key Biscayne) just off the exclusion zone and row in a small inflatable.
if there is no marking, i guess you can at least claim you didnt' know :-)
the Fish and Wildlife Commission is the primary regulatory agency in florida http://myfwc.com/boating/