Did Somebody Call My Name?
Don't "store" linens away for seasonal use per se, but do follow the age old French custom of placing newly laundered linens at the bottom of the a drawer or stack of things in cupboard so as to keep items in regular rotation.
At least once a year all linen drawers/presses are opened, contents removed, cleaned, aired,and lining paper replaced. Whilst linens are out quick inventory of items is made with an eye to what needs replacing, mending, reworking or simply taken out of regular use and a new purpose found.
Oh, also follow another old French custom by placing herb (mainly lavender) sachets in with linens. One day years ago took some old tulle and linen fabric that was lying about spare and ran up some empty packets. Filled with dried lavender leaves/flowers (puchased in bulk from France), then tied with a ribbon to close.
No musty scented linens here.
It may be worth one's time to empty all dresser drawers and take them out doors on a brisk day and let mother nature do the work. Sometimes a good airing can do wonders.