Bedsheets feel waxy/oily - how to get them "clean" again?

I've been struggling with a super annoying problem with 2 specific sets of duvet/comforter covers and pillowcases. No matter how I wash them, they come out feeling waxy/oily all over when they dry. I don't use fabric softener or dryer sheets. They are 100% cotton and around 3-5 years old. None of my other laundry has this problem, so I know that it's not the washer itself.

I have no idea how to get them to feel "clean" again, and it pains me, as they are excellent quality and a cute print design.

Do you guys have any suggestions on what to do? I looked up laundry stripping, but I don't have access to Borax (controlled substance in my country, Denmark), and people seem to disagree on whether it actually does anything.

Can you help? Thank you so much in advance!

Edit: I tried out soaking in my country's equivalent to Oxyclean, and IT WORKED! As soon as I felt the slickness of the water with the product in it, I knew you guys were on to something (remembering back from chemistry class that a base liquid fells this way). Thank you all so, so much for your contributions. They are all greatly appreciated. Now I can actually use my sheets again and don't have to buy new ones. Thank you!