The wet bandana will work, if it is wet enough.
I try to plant the mug around the heat source so that the heat is on 90% of the mug away from the handles. That way, the handles don't get too hot. Then, I have aluminum foil cut into a 1 inch wide strip and wound around the bottom of the handles. If the heat source goes directly onto that foil, it may melt. But, if the heat is going through the mug and is conducted out into the handles, the foil will not melt, and it gets no more than warm to grab by.
If you want to get crazy, there are silicone materials that are heat resistant and can be applied to the handles without adding too much weight.