I never like the idea of sitting on a long flight in a dirty, stinky, post-orienteering condition, so I usually try to find a shower option. I called this place a few weeks ago and they offer a day pass for $10:
Towne Fitness Club
124-C Main St, LaGrange GA 30240
Open on Sundays, 1pm - 6pm.
BYO lock; they have a towel service.