I came up with a cheap fix for keeping the meat out of the water.
I use 165 qt. ice chests, but this works with any large ones.
I went to a home improvement type store and purchased vinyl rain gutter downspout (rectangular tubes). I cut them to match the length of the ice chest interior. And lay them flat in the bottom.
Bag the quarters in garbage bags (keep the game bag on, if desired, and drop them into the plastic bag). Now, before everyone reacts to the plastic bag issue, just keep in mind that this works without trouble.
Lay the bagged meat in on top of the tubes, keeping the opening end of the bags up high in the ice chest.
Cover with ice to just shy of the top.
As the ice melts, it drains across the bags (a most efficient cooling technique) and collects in the bottom, in the tubes that you cut. Drain the chest every stop and top off ice during your trip home.
Meat comes home perfectly cool/cold and stays out of the water/slosh. It doesn't freeze up (which makes butchering a longer/tougher job).
For really hot trips across the western high deserts, we cover the ice chest with a moving blanket (or sleeping bag), even if inside a trailer.
Ice lasts longer than you'd think.