Old water treatment system causing mildew??
Their clothes and towels all mildew within 18 hours regardless of the heat and humidity and it leads me to wonder if it is either something in the washing machine , or coming from the water treament system.
Before going into speculative ideas about the plumbing, think simple, first. Mildew in clothes can occur if you leave them in the washing machine and do not dry them immediately, or if they are left wet for extended periods anywhere. Also, mildew sometimes/often does not wash out fully, so the clothes might have mildewed once and it never went away. The smell might just be covered up by laundry perfumes only to return a few days later when the perfumes wear out. Use hot water several times (with bleach if possible) to fully wash it out.
Also, the humidity of the storage closet is a more likely problem. The mildew smell might permeate the clothes due to mildew in the walls of the closet. Bleach the closets and use dehumidifiers to dry up the air. If you cannot afford the space/money for an electric dehumidifier, then there are products available for closets that chemically dry them out, using compounds similar to the little packets you see in new clothing/shoe boxes that absorb water.