I commute by bike daily in Austin, TX. In the summer I bring a change of clothes and wear a damp athletic shirt and damp headwrap. My commute is relatively short as well. I don't shower when I get to work and most of the time I smell perfectly fine. (Confirmed by people who have no problem contradicting me.) Yes, you'll sweat, but nowhere near as much as most people think you will if you plan it well.
Humidity would make this more difficult. Austin's not as humid as Houston, but it's not a desert either.
That explains. Contrast the climates of Georgia or Florida.
During the summers here, just standing outdooors in the sunlight wearing anything heavier than beachwear, is inadvisable at best.