Leaving the water in the cooler gives the entire cooler a greater mass and therefore it will change temperature more slowly. Which is what you want.
Water has a pretty high heat capacity---takes a fair amount of heat input to change the temperature of a large quantity of water by even 1 degree. So leaving the water in place will let the cooler absorb more heat before it changes temperature.
The best thing to do is start with your ice frozen in containers---anything from a Rubbermaid food container (of any size) or water bottles---whatever fits your cooler. That gives you large chunks of ice which melt more slowly, and contains the water so you don't have waterlogged food.
If you are buying ice, put it in some closed containers with a good seal. Double ziploc bags work well. That will keep the water contained.
