You might want try using a lavender scented fabric softener in the rinse cycle. At worst - you'd end up with lavender scented mildew.
How about putting the clothes out in the sun for 72 hours? Might sound crazy, but why is there a detergent called sunlight?
Thanks for the suggestions -- I appreciate it.
@michael2015: Although I didn't mention it in my original post, I really am aiming to eliminate the problem at the root. This is in part because I assume it's caused by a living organism that will continue to multiply if left unchecked, and also because it seems to be causing an allergic reaction (runny nose, sneezing) which probably would not be helped by dealing with the smell alone.
@dolphin: Yes, I did try exposure to sunlight, but without success.
I similarly have allergies to mold and mildew.
You should note - mold and mildew is a generic term - so frequently western solutions don't work particularly well domestically or locally.
If you can converse with neighbors - ask them how they resolve this issue. Personally - if the hand wash, soak, and then machine wash technique doesn't work - unless I have an unusual sentimental attachment to the clothes or towels - they get tossed.
As mentioned - vinegar as a natural solution (many flavors - for different natural scents), bleach (for whites), colorfast bleach for colored clothes, and borax as the ultimate killer. Borax is a typical cleansing ingredient used in many detergents. You can also try soaking in a solution of baking soda - but I can't remember if baking soda is also a bleaching agent. We usually just pull the latest box of baking soda from the fridge and toss it in the hand wash/soak cycle before we toss things in the machine - our washing machine is a front loader - which means low-water, so not conducive to soaking.
The best solution we've used to consistently eliminate the mold smell - hand wash your clothes and let them soak (baking soda) - then rinse thoroughly - then put them in the machine wash (with normal loads of detergent and tumble dry. If you don't have a machine dryer - you should expect that mildewy smell to be a regular thing with cottons and other absorbent clothing.
Alternatives to a dryer - a heater/dehumidifier in the bathroom - hang your towels, cottons, etc - just make sure you have a humidifier with some kind of oscillating fan and make sure you have air gaps between the laundry you hang to dry.
Is it dusty where you live ? Symptoms mentioned also fit those for dust/dust mite allergies. Which could be an additional problem.
I have sealed up the offending clothes in a couple of the big airtight plastic storage bags they sell at Walmart & Carrefour, which seems to have worked for controlling the allergies until my next round of cleaning efforts.
Soaking with baking soda sounds like a good idea -- I'll give that a try. As far as drying laundry, I fortunately haven't had mildew problems with clothes that I wear regularly, just with this batch that was in storage.
It's not especially dusty around my area -- I have lived here for a few years and no allergy problems until now
Just remember baking soda (I think) is actually somewhat abrasive - so soak, but don't scrub...