I sweat copiously and seem to attract road crud, dead bugs, sweaty salt, etc. It feels like sandpaper on my skin. I wear bike clothes and bring clean work attire in a pannier. Some days I'll even sink-wash the bike clothes and put a fan on them to be dry for the commute home.
When I get to work (early), I sit down outside and drink water until I quit sweating.
Then up to my office cubicle where I fetch my cleanup kit.
The ladies restroom is clean cool and very few people use it before work. I do a cold-water shampoo in the sink (fully-dressed), then filled a small bucket with clean cool water. Took all my stuff into the handicap stall, stripped, and used two washrags to get clean. One was a wet soapy rag using antibacterial liquid soap to wipe myself down with, head to toe, scrubbing areas if I was particularly road-splatted that day. Once I had soaped up and scrubbed, I used the clean rag to dip into the clean bucket to wipe the soapy water off my skin. Dried off with a XL camp towel (microfiber) and used a sham-wow to help dry and fluff my very short hair so it wasn't even noticeable it had been wet. Apply toiletries, dress, and clean the area and pack up. I felt very clean and refreshed.
It was important to leave the restroom as clean as it originally was - no hair in the sink, no splashes or wet floors.
I kept a couple pair of shoes at work, belts and some generic cardigan sweaters. No need for them to go back and forth on the bike.
Traveler-size bottles of shampoo, soap, chamois butter can be found and refilled from full-size bottles to fit in a quart-size zip-lock bag.
Despite my fears, this quick hobo cleanup was just as fast as a "real" shower where I'd be lingering wasting hot water instead of just getting the job done. Also much nicer than having to trot a mile or so to the "wellness showers" at the far end of the job campus. I usually got to work 45 minutes early so I had time to make my espresso latte and coffee and put up my stuff to dry before my coworkers showed up. And in the eveing, I would work a little late to miss rush-hour traffic and get a cooler ride home that summer.
About hair and stuff - I have a short carefree haircut. I figure as long as it's clean and well-trimmed, it will look better than the look I have when I'm totally out of shape and crabby not having my exercise. Safety and functionality comes before fashion. That's the fun part of being in my 50s - I still know what other people think, but I don't necessarily care.
Going slow to avoid sweat doesn't work. That just means less airflow and more time in the sun. Temps > 100F are common in the summer. Mornings are only 80F but humidity is about 80%.
And yes, there is a Women's Only Forum. Haven't passed anything gender-specific on in this post, but in the future you may wishto request privs from the mods.
YMMV