Do you put the meat you purchase in the grocery store  in cold water when putting it in you refrigerator? Why would you want to put quality venison in water?!
    If it was not gut shot,  properly field dressed, packed with ice in the cavity to cool if down ASAP, then processed into quarters once it was cooled down, it could be placed in plastic bags (don't seal completely in case the meat is not cold all the way through, this will prevent moisture/ mold from forming in the bags), then place it in the cooler in the shade with: first choice: dry ice - second choice: frozen gel packs - third choice: block ice - last choice: cube ice. NO WATER
    Invest in a digital thermometer and check it daily and make sure it is maintaining an internal temperature in the quarters of 40F or less.
Under these conditions you should be able to get 3 - 5 days in my opinion. If you cannot maintain  the internal temperature of the quarters under 40 they could start to spoil in as little time as 24 - 36 hours.
    If put in water the meat should start to turn white in color within a day or two, loose most of it's natural juices, and sacrifice it's quality even if it doesn't spoil.
    The absolute best choice is to quarter it and find someplace to have it frozen until you're ready to head home. This would cost less than you license, rifle or bow, and the other expenses of the hunt while maintaining the integrity of the meat.

Dave