I have holy books, eg: Rambam’s Mishnah Torah, Tehilim, Siddur, Chumash, etc. along with Tefillin in the closet and drawers in the bedroom that I sleep in. Is it ok or should they be double covered?

If one is sleeping alone in one's bed, the Sefarim need not be covered. If it is the bedroom of husband and wife, a single covering (eg in a drawer or closed closet) is sufficient. Double covering is required if a sefer has to be brought into a bathroom (for instance if it is in a bag in an airport).