One method of cooling your meat before the trip home we've used on remote summer hunts in Alaska is to put the meat in heavy duty plastic bags and seal them good and put the animal in a stream or lake or snow if it's available. Be sure not to allow water into the bag as this introduces bacteria to your meat.This gets the core temperature down good before you transport the meat. Unless you're in extremely hot temps, this should keep it until you can get it home. Once cooled, get it out of the plastic bags and into game bags to get good air circulation.
In Washington, you're usually not so far from a game locker that you can't get an animal there quickly enough to get it in a cooler without ruining the meat. Check out the area you're hunting in beforehand to see what's available.
But if you're hunting within a couple hours of home, I'd just get it there as fast as I could.