I grew up in Oklahoma, but in the humid north east part (north of Tulsa). These deals down work there - just makes it than much more humid. But out in central-west Oklahoma (like Enid area, on west) they work very well.
On a trip out to California in around 1960, my Dad bought one for the 53 DeSoto. It was not the window mount type, but sat on the hump in the front seat area, and was powered through the cigarette lighter plug. I don't think it was ever necessary to put ice in it, just water work fine, going through the desert on west.
(In the old days in Western OK, they would hang wet sheets in the windows - same effect, and it also caught a lot of the dust that was in the wind.)