Your assessment is flawed. Warmer temperatures do mean more water can evaporate, but that does not mean it will precipitate in nearby regions. There are many regions around the world that are hot, humid, and still dry as a bone (Somalia, Northwestern Peru, most Middle Eastern countries that border the ocean, etc.).This is like the other bad science assumption often tossed around by deniers: " Well if there is more water vapor then there will be more clouds and so the world will cool down!". No, it doesn't work like that.
There are conditions that need to be met for cloud formation and precipitation. If the atmosphere is stable, then it really doesn't matter how much moisture is present. If a blocking ridge forms over the region, then those warm moist air masses are going to move somewhere else. If there is a thick enough layer of dry air beneath the moist air, then it'll just be virga. If the air masses destabilize before coming ashore, then it'll just dump rain back into the ocean.
But I'm sure you know all this.