Folks above have covered most of what we know of the underdark of Athas. There was also a speculation thread on the Athasian underdark along with various ideas put together on this old thread. As for what became of Athas’ water there’s a lot of speculation about that here.
An old theory I had when I first started Dark Sun with my friends was that the water was somehow trapped under the Sea of Silt. There’s been mentions of habitable islands around the Sea of Silt where water spills free and creates small lakes and habitable isles in otherwise completely inhospitable lands. These days I suspect that isn’t where most of the water went, but I am positive there’s quite a bit under the silt.