It would cost a lot more to fix a shorted-out air handler than it would to install a thermostat (which you can always return) ...

and cheap window AC units are flying out of the local big-box stores. Not all of these discount units undergo rigorous QC before sailing across the Pacific, though, and a few wonky thermostats ...

Air conditioners can only really cool your home to 15 to 20 degrees below the outdoor temperature, so setting the thermostat too low can result in your system running without being effective.