People have learned this lesson the hard way with pets for many years. Your dog will just die on a hot day if you don't provide more shade and water than you need, because they just can't dump heat effectively.
I thought dogs didn't sweat. Maybe that's why, because it would be so ineffective with fur.
Basically, yes. Dogs have glands that produce oils and all sorts of stuff, but do not produce or use sweat for cooling, and being fully covered in fur you are correct that it would be mostly useless.
There are technically 3 types of sweat gland, and only one type is generally useful for cooling (there are always exceptions in biology).
Milk-producing glands (mammaries) are actually highly modified sweat glands!
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