How can I keep a cooler cold for a week without dry ice?

Use block ice instead of cubed ice — it melts slower and won’t soak everything. Make your own by freezing gallon jugs or large containers. Pre-chill the cooler and food beforehand, use a rotomolded cooler (like Yeti or RTIC), keep it in the shade, open it rarely, and consider separate coolers for drinks and food. With that, you’ll still have ice on day seven.