""I think their argument was that the air you inhale would have more moisture.""
Of course, simply wrapping your face with a piece of cloth won't inherently add moisture to the air one is breathing. If the piece was saturated with water, it could feel cooler, perhaps, leading to the impression of staying more hydrated; I suppose that if the wet bandanna actually cooled you off a bit, you might sweat less and save a bit of water. Ironically, if that were true, it might take water to save water? But that also makes me think of those humid conditions in the South - if the bandanna was so wet that it prevented evaporation, would it still feel cooler? But then again, if it were that hot and dry outside, the bandanna wouldn't stay wet very long anyway without constant soaking, adding to the irony!