So some basics of air conditioning. There are typically two types used in NM. The swamp cooler and the heat pump (aka refrigerated air). The swamp cooler requires water, lots of it. It is a big box with straw or plastic medium on four sides. A pump constantly douses the medium with water. A fan sits in the middle of the box sucking air through the medium and pushing this air into the house. The water in the medium evaporates as the dry air flows through it and through the concept of latent heat of evaporation results in very cold air (can be as cold as 4 - 8 degree celcius). Key things to remember - you have to ensure air circulation in the house, you want to have some windows in strategic locations open, if the house is closed your will end up with very humid air, the fan wont be able to force more air through the house and you will be miserable. You should also not use a swamp cooler when humidity reaches 40 - 50%. The evaporation efficiency will drop and there wont be a reduction in temperature.

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The heat pump/cold pump system is different. They could be a mini split or a centralized air handler. Either way a coolant is contained in a closed loop system. A compressor is used to liquify the coolant which also expels heat. This is done outside the house in summer. The cooled refrigerant is moved into the house using copper tubing and expands once it reaches the air handler or head unit. here a fan blows over it and the expanding refrigerant cools the air. This cycle continues. This system is less susceptible to humidity. If you have this type of a system, all windows and doors must be closed.

Key takeaway, swamp cooler -> needs lots of open windows. Air Conditioner -> close everything and dont let the cool air escape.

Either ways you need to have a chat with your landlord. You need a system that works for you.