Just plug the drain.

Plug the drain from the inside with a rubber stopper or cork, then seal the outside with duct tape. That’s it.

Most cooler drain plugs are threaded plastic that can vibrate loose or leak under motion. A cork or rubber stopper (hardware store, couple bucks) creates a tight seal. The tape is backup insurance — wrap it around the outside of the drain fitting. Stretch it tight.

If you don’t have a stopper, a piece of plastic wrap under the original plug helps, but it’s not as reliable. The cork trick has never failed me across multiple cross-country drives.

And if you can drain the water entirely before driving, do that instead. Ice stays cold enough for a few hours without water sloshing around. But if you need the water (e.g., kayaking trip, camping), seal it properly.

Future you will appreciate not cleaning up a wet trunk.