I've done cold water sink shampoos then did a head-to-toe wipedown in the handicapped stall with a washrag with a smidgen of diluted liquid antibacterial soap, then another wipedown with a wet clean washrag.
Followup with anti-perspirant and clean clothes.
It is as effective and as quick as a shower, although it is not as mood-enhancing.
Be neat, don't leave drips. Use sampler-size bottles for soap/shampoo/etc.
I got a XL camp towel to use. It is very thin, packs small, dries fast.