Stacey,
I have learned the hard way that unscented and hypoallergenic products can indeed be neither. Many unscented products actually contain masking fragrances to make the products smell unscented! These fragrances are made from petrochemicals. This is the problem! Aveeno is a nice idea. However, this product line also contains petrochemicals. I'll bet that if you go to a Trader Joe's or a health food store and get some plant based detergent and softener, your problem will be solved. Bio-kleen and 7th Generation are both good brands. These products are more expensive, but we can't use anything else. Oatmeal soap is another good idea--if you get natural oatmeal soap. This should be easy to find, as well. If you read (between the lines) lots of labels, you will be able to really know what you need and also what you are buying.
By all means, continue to breastfeed. Also try a natural calendula cream for rashes. (Vaseline is not natural.) We love Hyland's brand. Weleda makes some good products, too.
You will be able to solve this problem.
Amy
PS Deborah is right.