The Dead Sea in Jordan includes the eastern shore of that body of water.
South to Aqaba
To go south, just stand directly on the main highway and signal. The same highway proceeds past the Dead Sea to the potash town of Safi and through the desert to the Red Sea.
Camping on the beach is quite convenient. However, be aware that even in winter mosquitos can be quite fierce, and the clay-like surface of some beaches is liable to stain your belongings if you don't put a piece of plastic underneath.