You have my sympathy. Inflexible RTO has delivered a big ball of suck to people in the DC area (and other places with godawful traffic and ridiculous commutes). So much productive time is lost sitting on the roads/dealing with Metro.
On days when I have to look like a grownup after my ride, I do a lot of what others have suggested. I start with clean hands — cleaning tack with a sponge, hot water and glycerine soap helps there! Even gets the crud out from under my nails. Then I wipe arms, neck and face with a wet bandanna or towel to get the first layer of dirt and hair off exposed skin. Shuck off barn clothes (including skivvies) and use baby wipes (cheap unscented store brand) for a showerless shower. Reapply deodorant, get dressed, apply makeup. I bring a go bag with me, but if this was a regular thing I’d just keep duplicate grooming items at the barn.
I am not a hair person, so I just brush and smooth into a low pony or bun, or rinse under hose and finger comb when I’m in my bob era. But I imagine dry shampoo might help if you want to have some semblance of style without grease.
Finally, I blast the a/c for the first five minutes in the car. This helps reset my body, with its temperamental middle-aged thermostat, from “outdoor heathen mode” to “indoor propriety mode” and helps keep me from sweating away all my hard work.
If I vacuumed my car more often, I wouldn’t have to worry about my clean clothes picking up random grey hairs. (His, not mine. Well, OK, mine too. Sigh.)