If you can't find dry ice, use bags of ice you can buy at most 7-11s, convenient stores, and even gas stations. I carry a box of the 2-quart ziplock baggies, and put about 10 lbs of ice in each of them, and put them inside the chest cavity, then put the cheeseclothed carcass in plastic garbage bags, more bags of ice on top of the bags, and then a thick wool blanket on top of everything for transportation. If I had large enough coolers, I would skip the plastic bags, and put the meat in them, I still would cover the cooers with the blanket, and leave a bag or two of ice on top of each of the coolers. Cold air goes down, and that ice can keep your cooler cool for many more hours, while you are making a long drive home. If the cooler stays cold, the ice inside also stays cold, and you don't have to replace it as often, if at all.
Just a comment about hanging meat: When you get above 5,000 feet in elevation, the air is thinner, it gets really cold at night, and there just is not much moisture in the air. Bacteria needs food( your deer), moisture( the meat again, but also a lot comes from the air), and temperatures above 40 degrees to work. If the air temperature drops below 40 degrees at night, it kills the bacteria in the air! The reason you can get away with hanging carcasses from a tree in the high country is because it is mid day before the bacteria can reproduce in any numbers that they can drift through the air and find you deer to feed on. With the limited moisture in the air, the air is mostly sterile. Add to that the sun baking the surfaces of the rocks so hot by 3 p.M. that you can cook eggs on them, and bacteria are literally fried and killed during the heat of the day, too.
Don't do those those same things if you are hunting down here on the flats, below 2,000 feet. We have very high relative humidity, and the air is heavy with moisture that makes a perfect place for bacteria to grow, breed, and multiply. Our nights don't get so cold, unless you are in the upper midwest, where the fact that the temperature can stay below 40 degrees most of the day, if not all the days you are hunting makes you fairly safe. However, you generally are not hunting miles from home, or from a meat locker here, either. I do like the idea of using vinegar to remove the blood and dry the carcass after field dressing. I am going to try that with my next kill.