I commuted for about two years, 10 miles each way, to a place without a shower. A few things make it possible:
(1) Shower before you leave home.
(2) Go slow in the mornings. (In the summer, balance speed and sweating as best you can.)
(3) Check email until you stop sweating.
(4) Use unscented baby wipes in the handicapped stall of the restroom to wipe down before dressing.
FWIW, I carry clothes (in panniers) daily, except I'll park shoes behind a computer in my cube. The key is to wash before you leave, so there's not an accumulation of stench when you arrive, dry and wipe when you get there, so the sweat doesn't "ferment." Plain sweat doesn't stink until it's fed skin bacteria for a while. Even after I'd been doing that for a while, there were people who were surprised that I did not shower after the ride -- those who hadn't seen me over the summer and didn't know the routine were amazed I didn't smell like a jock or a locker room.
Also FWIW, I prefer a shower at work.