Most modern ACs in fact dehumidify and cool in one step with a compressor and then use the collected water to help carry heat away from the compressor by letting the water evaporate into the exhaust to save power (it's why you rarely have to empty water from modern dehumidifying ACs).
Someone could make a “Einstein dehumidifier” using 19th century technology and roll it out across the world. If embedded in buildings it could cool them constantly with no electricity required.
Anyway, the best way to learn more about this is to search for resources describing the "psychrometrics of evaporative cooling". Psychrometrics is the study of the thermodynamics of air and water vapor mixtures. Learning about the psychrometric chart, in particular, is helpful since it is a very clear, expressive tool that will illustrates how adiabitic (no change in thermal energy transfer) cooling can occur by adding humidity to dry air.
I think this entry from engineering toolbox provides a good start: https://www.engineeringtoolbox.com/evaporative-cooling-d_698...