We say that in NY too :)
NJ is south.
Lower 48ers for most of you!
The meaning of Going down the shore depends on from which sector of NJ you hail; that's why Darryl Hannah's backhanded comment about Wildwood in the movie Wall Street was clearly the result of an uneducated dolt: some NYC thing like her character would have instead spoke of Mannesquan or Long Branch or Asbury Park.
FWIW, the best cheesesteak I ever had was on the boardwalk in Wildwood while the best ice cream was from a shack in Asbury Park near The Stone Pony.
And for the record, I hate Springsteen.