While reading all the answers here I remembered reading on another site about getting smells out of bottles and food containers. People were saying to put scrunched up newspaper in them then put the lid on and leave them for a few days and all odors were gone.
I was thinking this may be a way to remove the odor from fabric. Place in a bucket with a lid or place in a plastic bag and tie the top. Worth a try. If it will remove odors from plastic containers and salsa bottles it should work for fabric.

My belief is; If it isn't something I would give my grandchild I won't give it to someone else's child

Maybe if someone in your group can find a better place to store fabrics, a spare closet in someones home? Basements and garages get musty and pick up odors. I know closet space is hard to come by though. Maybe an attic would be less musty?
Sunlight and air might help but hopefully there is no mildew on fabric.

I had fabric like this, too, donated to use by a church for charity quilts. It had been stored in a plastic box in a room with a dirt floor. I washed with unscented Tide, Oxyclean, used vinegar, rewashed, put it outside for air...thought it was okay. THEN I ironed it...OH MY! That smell just hit my nose. BIG TIME. So it was a redo of all the washing and so on...only to have the smell still there when I ironed it. So that was it for me. I now use leftover fabric that I bought and use that.
Sometimes fabric just can't be saved. :(

Anything made of fabric for donation should always be washed in hypoallergenic detergent and no fabric softeners or dryer sheets BEFORE donating or probably even gifting! Just because I can't smell mustiness or chemicals doesn't mean they aren't there.

I would hang the fabric outside even if it is winter. I remember my grandmother hanging laundry outside even in the winter. The sheets and towels came in stiff as a board BUT they had a great smell. She would then hang upstairs in a spare room until they were dry to use. Also they were really soft. She didn't use anything except laundry soap in the wash. I can't remember what the soap was. I also remember helping her with the old ringer washer that was in the back "outdoor" room (she called it) where the sistern was. It would get chilly but never really cold in there. The great outdoors can cure alot of things. Sorry, got off the track a bit. I haven't thought of her and the farm in a long time. Thank you.

Also when storing fabric try using the vacuum bags. They are air tight and neither water or smells can get in. Then you can store them where ever you want without fear. I mean the storage bags that you vacuum out the air and seal them.

I am not a fan of changing one odor for another. Even if it is "fresh" to me, it is still an odor. ODOBAN =Original Eucalyptus Scent

On the last page, beaqltr says to use bleach. Yes, chlorine bleach. Add it diluted to the machine. With a front loader, wait till the water is all in, then pause and pour in the solution. You can do this until the "add garment" light goes out. It will take away the musty smell.
I've been there, done that with the hot iron on musty stuff, but the chlorine bleach took it away completely. A quarter cup of bleach in a pint of water should do the trick. And it doesn't really fade much either. I couldn't notice any fading.

Try baking soda in the wash. That usually will clear out odors. Why throw away good fabric when it can be deodorized? May take a few washes, but should come out 'fresh' after that.

I process all the fabric donated to our local chapter of Project Linus, and I have also received fabric that smelled old. I have tried all the suggestions mentioned here (and have had great success with Odoban). If nothing works and I can still smell the "old" smell, I don't use it. It's sad, but I don't think it's worth going through all the effort to make quilts you don't feel completely good about giving away.

Makes me realize how important it is to use up stash and not let it sit for decades!