So here’s the broad brushstrokes of why a small evaporative cooler is probably going to make things worse rather than better with regards to cooling:

Water absorbs a lot of heat and it does it far better than air does. Hence why air is often used as insulation - if it’s held in place like with bubble wrap or double-glazed windows it works remarkably well.

That changes as you increase the amount of water in the air, which is why saunas feel hotter than the equivalent temperature in drier air.

And this is the principle that evaporative cooling works - including very old technology such as Coolgardie safes, yakhchals, and even biological systems that we use like sweating or panting. What happens is heat is absorbed by the water and then that water is carried away into the air, especially with airflow, thus making the wet or damp object cooler by dumping heat into water and the water being dumped into the air.

This is also why it’s more difficult to stay cool in higher humidity environments - your body’s ability to dump heat into sweat and have that sweat evaporate is reduced as the air becomes more saturated with humidity, to the point that in very high humidity conditions your body is less able or perhaps even completely unable to shed excess heat.

So a household evaporative cooler works in a similar fashion - it creates a lot of humidity in the air in your house, the ambient heat is dumped into the water in the air and, this is the crucial part, that air is then pushed out of the house into the atmosphere, thus carrying the moisture and the absorbed heat with it. That’s why you need to let air escape when using evaporative cooling.

Where this principle falls down or even becomes counterproductive is if you’re trying to cool yourself down with a small fan in a room or a house and you’re dumping a lot of moisture into the air but you aren’t able to push that air out into the atmosphere; instead of shedding heat, what you are doing is creating a more sauna-like or greenhouse-like environment that absorbs heat and holds it in the air so any heat sinks in your house like a warm brick wall are going to distribute that heat into the air, with a potential to encourage that wall to start absorbing and dumping more heat into the air if it cools down significantly enough.

Thus in one respect you have a problem of creating air that is going to hold more heat than it already does. But that’s only half of the picture.

The other half, and this is at least as important, is that you are going to impede your body’s ability to cool itself down the more effective your small evaporative cooling system works because for every percentage higher the local humidity is, the less your body is able to do its own evaporative cooling via sweating. So the air itself gets hotter and is much better at holding that heat but also it makes your body “better” at holding its heat as well.

So no bueno.

This is not the case if you’re outdoors or if you are okay with dumping the excess humidity and heat into other rooms in the house or if you have a complicated setup which is also drawing or pushing air out of a window into the atmosphere but at that point you’re probably using so many watt hours that a heat pump/reverse cycle air conditioner would be more efficient for cooling. If there happens to be a stiff breeze that is steady and travelling perpendicular to your window then it will probably carry the humid air of your room quite well but counting on it being strong, steady, and constantly in just the right direction is pretty damn unlikely so it’s not something I would count on at all.

What you can do to leverage evaporative cooling effectively is to have a strong fan, as mentioned above, and to create more opportunities for evaporation from your skin, such as using a spray bottle on your skin directly or wearing lightweight clothes like sarong or keffiyeh that are nice and damp. Yes, you’re still dumping humidity into the local air by doing so but you’re going to get the most bang for your buck this way and you aren’t likely to significantly increase the humidity in the air, especially if the room you’re in is open to other places in the house. Of course, as mentioned before, as the air gets more humid it becomes harder for evaporative cooling to function and this applies to damp clothes too.

I could go on about other stuff like how those dinky and DIY cooling projects that use ice cubes are almost certainly counterproductive too but I’ll spare you that particular lecture - just apply ice directly to your skin (wrapped in a wet towel around your neck or on your face, in ice baths even if it’s just your feet, or ideally consuming ice directly.) Get yourself a knock-off yeti cup or Stanley cup, fill it with ice, top it off with water and sip on it through the day and you’ll get the most out of the cooling that ice provides.