> higher temperatures drives increased evaporation which leads to increased droughts
Certain types of vegetation work better in higher temperatures. Remove vegetation altogether, and the combination of higher temperatures, wind, and rain will transform the land into deserts. Preserve old-growth forests, and they will remain lush, green, and wet regardless of temperature variations.
> the evaporated water in the atmosphere has to come back down again, which it will tend to do in a great big splash
Atmospheric rivers are interesting.
https://en.wikipedia.org/wiki/Atmospheric_river
> don’t really penetrate deep enough into the soil to alleviate the increased evaporation (too much too fast)
That depends again largely on the type of vegetation growing there. Forests and wetlands can capture the rain and allow it to seep, while pastures and corn fields will retain very little, which can lead to soil erosion and droughts / water shortages.