Assateague Island is about 30 miles from my home.
Perhaps, familiarity breeds contempt. In any event, driving on the beach, at Assateague, necessitates purchasing a $70 one year permit, a $10 entrance fee and possession of required equipment, to drive about 10 miles of unchallenging seashore.
See Assateague OSV Use
The US Park Rangers enforce all rules ( permit, equipment, speed, DUI, etc. )
veryagressively.
I'm sorry to say there are really no good off-road areas, on the Eastern Shore of Delmarva. The closest place, I know of, is Paragon Adventure Park, in PA, about 4 hours away.