I also usually stick the container in the cooler, but I usually have things sitting on top of the ice in plastic bags so the food and drinks are also separate from any fish I keep. Any fish going into the cooler are killed and bled first so no thrashing around once they go in.
Some days, I’ll just stick a ice pack into a insulated lunch bag and keep them in there. I use the same bag for ice fishing, only I use a hand warmer instead of ice pack.