No, you were correct initially. It's just a matter of there being two different kinds of cooling systems. The cheap kind are just fans that blow across the surface of the water. They work through evaporative cooling since the air movement will increase evap rates and the process of evaporation removes heat from the system. This will only cool the water by a few degrees though. The really expensive kinds are the fancy ones that use actual refrigerant. They pull the water from the tank, cool it, and then return it.
Regarding the topic of the thread, it probably is some sort of disease going through your population, although I'd also recommend finding a different source of fish once the one month period is up. Some shops just have consistently bad fish, and if you keep purchasing from them, you'll just keep running into the same problems.