Wash them in the wash machine. The drier is what _kills_ them, the bill will mostyly hold up to the initial getting wet in the washer, but once soaked, it can't take the beating in the drier messes up the cardboard.
So air dry.
There is a plastic cage for them, got one at Harbor Frieght, yshaped like a cap, you put the cap in the cage & throw in the washer, keeps them in shape better. It does seem to help.