There's really no law that says "you must perform complete ablutions after the slightest hint of exercise."
We do have showers here... but they don't seem to have hot water when I get to them, so I gave up on them. So, like the other folks have said, I piddle around for ten minutes (or an hour... depending...) and then duck into the ladies room with my change of clothes soap and washcloth and do a wipedown of the critical spots.
I don't sweat a whole lot anyway... and I try not to work too hard (but that doesn't always work). I'm afraid I've never been Hygiene Queen...
And with my Xtracycle, yea, I can carry a week's worth of clothes in if I want to on any given day. Heck, I could put a garment bag in the back. *That's* not a problem :-)