Inhaling Fibers from Bedsheets

I have dark navy cotton bedsheets and over the past year I've noticed that the sheets "pill" or "shed" small fibers. These look like the typical tiny little microfibers that make up the dust everywhere indoors.

Recently I noticed that they leave residue on my hair. I've had a bit of a cough, and I noticed that - sorry for the graphic image - when I blow my nose I can see the tiny microfibers in the tissues if I look closely.

This is concerning because it means that these fibers are in my lungs. Also shows that dangers can be lurking where we least expect them. Has anyone experienced this, and can anyone recommend "safe" hypoallergenic bedsheets?