They do need to be washed - head gear can get soiled, smelly and harbour bacteria enough to cause rashes or even open sores. Gentle dunking in warm water with a drop of detergent or even a rub with hand soap will clean off oil and remove salts - though if the soiling is heavy it might need to soak a while. Several rinses with a splosh of white vinegar in the last one (to give it a pH where wool is happiest) should restore the cap. Squeeze it in a towel to remove water then stretch it out into shape and let it dry in a warm place.