You have a cap on your truck?
I would use a bag of pellets sold to put in horse stalls for bedding. The pellets would absorb a LOT of moisture and your broilers would arrive cleaner for butchering.
You can clean this bedding out of the truck bed and put it in your garden or compost pile.
See if you can schedule your broilers for EARLY morning butchering. Then you can load them way before dawn and haul them before the sun is really hot.
Remove their food at noon the day before butchering. Digesting food causes the body to produce heat.
What do you use for bedding to haul them on?