I would leave them as-is since you want to store bedding and linens.

Bedding can hang on clothes hangers quite nicely, that's how it is sold in second hand stores and it stays very tidy and easy to see. My mom always hangs her long table clothes. (I fold most of mine!)

The shelves on the sides look perfect for folded towels, pillows, washcloths in baskets, etc. If the shelves are splintery just lay down shelf paper first! (NOT adhesive contact paper.)

Old cedar tends to be dried out on the surface, so a light sanding is advisable to reactivate the oils. You can also apply cedar oil directly to the wood.

sanding is the only thing to do. it will bring back the cedar smell. never put a finish on cedar it totally negates the reason for having a cedar lined closet