Does the cover have a tag with how to wash instructions? I want to say some of the water proof outdoor rated covers have specific instructions on how to clean them so as to not mess up the waterproofing feature.

If it's just dusty I would grab a helper and shake it out really well, and/or vacuum it.

Drying it could be done on a hot summer day just by putting it outside spread over some lawn furniture, etc. to keep it off the ground.