My thoughts, take em or not. :)
1. Do not use fabric softener or dryer sheets. If you want something, put some white vinegar in the washer.
2. Avoid perfumes. OP, you mention allergy issues, and for some people, perfumes are a real problem. If they're not for you - great. You're just helping to not make someone else miserable. Laundry, and soaps can really be an issue because if you're sensitive, you can't escape from your clothes. (Or hands. It's really great being temporarily allergic to your hands)
3. If you live in a region which is typically humid/damp, line drying may simply not work. If that's the case, mold is likely more common in general and is a common allergen.
4. Regular cleaning of the house is going to help a lot more with reducing dust, etc than not line drying would. It's also more likely to make the house not smell bad than air fresheners.