Get some trial size shampoo and refill every so often from a normal size bottle kept at home. Head & Shoulders smells nice, rinses clean.
Use the shampoo as soap or also decant liquid soap (eg: dial anti-bacteria) into a small travel-size bottle.
Carry it back and forth in a freezer (heavyweight) quart zip-lock bag along with comb, antiperspirant, whatever.
Just set the ziplok on the floor, throw the towel over the outer shower curtain rod.
Shower feet funk are not an issue at my workplace, understand not wanting to carry flipflops / pannier space constraints. Carry some alcohol-based hand sanitizer and clean your feet after you thoroughly dry your toes. Also comes in refillable trial-size bottles.

Get a XL thin travel towel - they dry fast and pack small. Either keep it at work if your cleanup is near your desk; or carry them on the bike each day.

Washrags are a must if you don't have a shower. One for soap, the other for rinse. Used a small bucket for clean rinse water that doubled as a tote/storage for toiletries whem I kept them at the office.