How do I get sheets smelling fresh?

We only have two sets of sheets which we trade off while the other is being washed.

But after the first few months they start to smell musty, as if they never got clean.

I use Tide free&clear in an automatic dispenser set to “low” in warm water and don’t use fabric softener of any kind. I’ve tried adding a cup of vinegar to the rinse cycle but it doesn’t do anything.