There’s very little access to water on Athas and those that have it, control it; so, how to people bathe in the Tyr-Region? Answer: they don’t, really. Unless the Sea of Silt has some sort of cleansing-ability, the people on Athas stink; BIG-TIME! You can probably smell a city before you see it.
And, not being accustom to water or bathing, your average Athasian wouldn’t think to clean themselves if they were near the Dragon’s Bowl or near a mountain-stream.
Historically speaking.. sand baths. They’ve been used for thousands of years. You use dry sands to scrub your skin. I’ve done it myself when overseas, it’s fairly effective. For body odor, it’s mostly from bacteria, in hot dry environments they bacteria doesn’t grow all that well, so much less odor. Historically the use of various herbs and oils along with the sand scrubbing removed most of the odor you’d expect in more humid areas. Sometimes wet sand was used for sensitive areas like armpits, groin, and feet. That water didn’t even have to be drinkable.
A lot of desert folk don’t sweat all that much, they tolerate heat a lot better then you may think. I’m sure Athasians with thousands of years of exposure to such hot environments have adapted fairly well to it. There also a gene (ABCC11) that most east asians have a variant of which makes sweat basically odorless.
Sand bathing makes sense.
Oil and something like Greek/Roman strigils might be used some places, too. There’s a reference in Amber Enchantress to some kind of oil being applied when wrestling; I don’t know if it’s just for slipperiness or if it’s a cleaning thing like Greco-Roman athletes did.