If we put water in an earthen pot then water is being absorbed and it reaches the surface of the pot and then water evaporates and makes inside water cool. So is it right that if we wet the earthen pot ourself then, will the process of cooling of water become shorter and water will cooled down faster?
If you put earthen pot in water before water inside pot reaches surface, then Yes, cooling will become faster.
For instance, your hot utensil cools when plunged in water.
But, final temperature of water inside pot is independent of above two ways of cooling.