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."

Two great suggestions above and I have used them both. The first being getting wrists in cold to cool water will indeed get your core temp down fast there is a lot of blood flow thru the arms and hands and it only takes a few minutes to really feel the heat being drawn out. The second something I have never tried while biking out west but all the restaurants out there that have outdoor seating have spray misters that amazingly cool you down. I have never seen a bike equipped with a mister but I bet it would work well. I have a camelback and never got that used to using it here but I’m betting filled with water and kept in the fridge overnight would be good to draw some heat out of the core and supply the extra H2O needs. Ice??? I have dumped water on my head or on a towel and put on neck to evaporate.