What is the difference between down the shore and at the beach? Does the latter imply you are physically on the beach?
This is a Jersey espression and refers to heading down south to any of the beach towns on the New Jersey shoreline.
Consider that "the shore" in NJ was largely developed as resort communities for Philadelphia and New York during the summer months: in fact some resorts were built and owned by the railroads which took people there.
The beach is only part of the experience which included other attractions. In addition, many people have and had summer homes there.
"The Shore" is more a region than a location.
My impression is that "down the shore" is what some people from New York/New Jersey/Philadelphia say to mean they're going to the Jersey Shore, which is south of New York. It was also the name of a sitcom set in Belmar, a town on the Jersey Shore. If you're in New York City, and you go to a beach on Long Island, that is not "down the shore".
The idiom in the mix here is to say "Down" instead of "On the shore" which is why it may sound strange, just as "down the town" should be "in the town". (And also the difference between beach and shore).
Here in Ireland it has resulted in an even more twisted geographical references, such as "Down the north" or "Up to Dublin" (Dublin being in the south or east).
"At the beach" would usually imply being at the sandy place where the sea meets land, "On the shore" would usually be used to differentiate with being "On the sea".
One could also possibly say "At the shore" which would indicate that one was at the point where sea meets land, irrespective of whether it was a beach or not, a beach generally being a gradient of loose particles such as sand or pebbles.
"down the ..." is just a New Jersey idiom. The reason that it is "the shore" instead of the beach is that the Jersey shore is primarily made up of barrier islands which have both a bayside and an oceanside (a beach) and other amenities like boardwalks, rides, bars, junk food, etc.