I think you can miss walking around Amman. If you can restructure, I agree the Dead Sea does not need that much time. Wadi Rum can be an easy day -- a jeep ride -- if you like, and you can see it without overexerting. you can also camp there, do an early morning balloon ride, etc, but that is not as restful.

Petra is a wonder of the world, and worth 2 days. I booked 3 nights there -- one for arrival from the drive down from Amman, visiting the next day, and then a "swing" day which was good as we got sick and used it (the plan had been to use it to see Wadi Rum). The drive south from Amman features many sights you will miss - the Moses Spring, Mt Nebo, and Chapel at Madaba ( I found the famous mosaic to be much smaller than exxpected and underwhelming).