It sounds like a decent option might be to cut out dairy and meat products until you can get a suitable cooling system? (aka: go vegan for a bit!)
Most (if not all) root veggies and fruits are perfectly fine with being set at room temperature. In fact, I keep all of my fruit in a bowl on the kitchen table and just grab a handful of berries or what-have-you when I feel the need. You can substitute milk for almond, soy, rice, or coconut milk which can be kept at room temp. In most recipes, eggs can be substituted with ground flax seeds. All of your grains, beans, and legumes can be room room temp as well as some meat replacement items (Tofu if packaged in a sealed container might not need to be refrigerated until opened).
Another solution might depend on where you live. If it's winter with an average temperature of around/below freezing, you might be able to store your dairy/animal products in a cooler outside. If you do this, I'd suggest you buy a thermometer to put inside the cooler so you can keep tabs on your food. I'd also suggest looking up the items you plan to store to see what temp they're best preserved at and for how long. (After a google search: The inside of the cooler should be no warmer than 4C/40F in temperature.) I've stored cheese, milk, and cold cuts for around 4 days in a cooler several times with no issues on camping trips, but raw meat products might go bad quicker.
You'll still have to go to the store more often like everyone else is suggesting, but it might cut the trips down a bit.
Hope this helps!