Best way to deep clean towels and linens?

I agree with all other responses about tracking down causes of odors but will also add line drying will eliminate odors from clothing permanently. Odors result from a bacteria buildup. Sunlight kills bacteria naturally, bleaches whites naturally, and maintains the integrity of clothing longer. Turn darks inside out. During allergy season do a finish dry in the dryer for 15-20 minutes after line drying. You’ll be amazed at the difference.

Determining if your dryer is the musty smell culprit is pretty easy. Wash identical items (e.g., t-shirts, socks). Line dry some and machine dry the rest. If they don't smell the same then most likely your dryer is contributing to the problem.

I can tell without this test since I machine dry white loads and line dry dark loads (the latter to reduce pigment fading). Dry cotton should smell like cotton. Dry linen should smell like linen.

Again, do not use dryer sheets which spread foreign substances all over the fabric molecules.

One other spot to check would be your dryer hose. A build up of lint will cause a bacterial overgrowth and a musty smell from mold and mildew. If your dryer is up against a wall and cramps the hose, this can happen easily. If your hose is who knows how old- it's probably full. Give your dryer a good vacuum, and consider cleaning or replacing the hose. The trouble may not be the towels, your detergeant, or your fabric softener. It may just be your dryer. If you aren't drying the clothes, then check your washer. A top load machine should be left open after use, to dry out completely. You can wipe down the inside with vinegar or bleach, as well as run an empty load with either product. If you have a softener dispenser, clean that. For front load machines, run a cleaning cycle with bleach, or one of the machine cleaners made for the purpose. Honestly, smelly "clean" laundry is usually a problem with one of the machines.

Along with what @702551 advises, wash in hot water and some Oxyclean or oxygen bleach. I like to use vinegar as a fabric softener in the final rinse. The vinegar is the only thing I have used that gets rid of cat urine. My laundry does not smell of vinegar either.

First of all, do not use scented detergent or scent additives. Many of these tend to attract oils which is the last thing you want to happen during laundry.

Same goes with fabric softeners. Don't use them.

Use unscented laundry detergent and add some borax.

After the cycle is complete, run a second cycle without laundry soap but add some baking soda (about 1/4-1/2 cup). Some do a pre-soak with a baking soda solution.

The water going into your washing machine is an important factor. I have great municipal tap water and I always cut back from the recommended soap quantities to 1/2 or even as little as 1/3. This is both for laundry and dish detergent.

You will need to experiment how much soap you need to use for your particular water.

Best of luck.