Instead of using regular ice, get some of the reusable/refreezable BlueIce packs. I use these every summer in my garage humidor (freezer portion of an old unplugged standard upright fridge/freezer). I have 4 of them, & I use 2 at a time. When the 1st pair is no longer cold, I replace with the frozen ones & put the warmer ones back in the 'real' freezer. Have I mentioned I live in AZ? Never had a beetle problem, keeps my stash below 75 degrees all summer long.
Helpful tips - Keep the packs in the upper portion (cooler air is heavier so flows downward). And put a dish towel under the packs so excess moisture doesn't mess up your sticks.
Of course, your other option is to start smokin double-time!