Pinworms do not reside in the mouth, and you need not fear that this is so. It sounds as though you are having a relatively common problem of dry mouth whereby the mucous lining of the cheeks begins to slough off and can be removed as white strings of mucous. A good treatment for dry mouth is a mouthwash/rinse called Biotene, which is available over the counter at the drugstore. It contains enzymes that are needed in the mouth to aid in the breakdown of food while chewing, and lubricants.
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