I am also going to suggest using vinegar to replace fabric softener. Both my husband and my daughter have multiple respiratory allergies, and I am always looking for a way to make their symptoms just a little better. Dropping fabric softener helped a little, and now I just pour straight vinegar in the liquid fabric softener dispenser in the washing machine. Works like a charm - no smell. It's also supposed to help with the build-up on the clothes - both from soap AND stuff in the water (which is really why I'm suggesting it - it's a happy side-effect in my case, but in your case, might be a main benefit!).
Incidentally, you might want to clean your washing machine, too. If for no other reason than the hard water can build up in there, too. You can use a vinegar/bleach solution in a super hot cycle (google it for the exact proportions), use NO SUGAR lemon koolaid or lemonaid packets(it's the citric acid that does it), or buy appliance cleaning packets (in our grocery store, it's in the dishwasher detergent aisle, but they work for washing machines, too). You can do the same for your dishwasher, too. (I prefer the cleaning packets, myself.)