No showers in our locker rooms either. I use a spritzer that I found on this wonderful forum, to wit:

In a spray bottle, mix equal parts wintergreen rubbing alcohol and witch hazel.
Carefully add several drops (depending on the size of your bottle) of rose and sandalwood essential oils. Season to taste - a little can go a long way!

I sweat *profusely* and have no problems with odor or discomfort after a 10 mile commute on a hot and humid day. Here's my routine:
Shower the night before or just before leaving.
I wear Target 'sport' underwear with Spandex under short running shorts, T-shirt (optional), short socks & shoes and a reflective vest. The idea is to have as much skin exposed as possible for cooling. The light clothes dry quickly and can be worn home.
I moderate my effort in a (possibly vain) attempt to sweat a little less. This means leaving earlier so I don't have to push to get there in time. I also build in extra time for cooling off when I get there. I keep an eye on my cyclocomputer to keep my speed down and coast whenever possible.
A couple of miles from work, I back off and try to start the cooling process. If it's really hot & humid, I'll stop by a grove of trees near the building and cool off in the shade for a bit.
Once in the building, I go to the locker room and splash cold water on my face to trigger the 'diving reflex'. This really speeds up the cool-off period for me.
I stand near an air vent and let the sweat evaporate. If I wipe it off now, I'll just get hot again.
Take off clothes, wipe down with a towel and liberally use the spritzer. The spritzer is very cooling; the alcohol and the witch hazel evaporate quickly to leave behind the essential oils to smell nice all day!
Put on the uniform and start makin' the doughnuts...

Hope this helps!