Not sure how OP's sweat glands work (especially in dry heat), but it usually takes me 15 minutes to stop sweating. Check your email, maybe get a cup of coffee, waste a bit of time until that happens. Then you can clean up. A bit east, if I time things just right, the last of the sweat keeps the salt dissolved so the baby wipes clean everything right off.

And +1 to "shower before you leave home."