I use various unscented detergents. Seventh Generation, Ecover Zero (this one says not for silk or wool, as it has enzymes to remove protein stains), Planet, or Vaska -- whichever is available / on sale at the store I happen to be at when I think of it. I just got some handwashing detergent by Planet.
I'm very sensitive to fragrances and some other chemicals but had never thought about preservatives in detergent - I know they're a problem for me in some face products and in contact lens solution. What was your allergy testing? Do you have contact allergies (skin rash) or other effects? I get headaches, itchy eyes, or upset stomach, and sometimes I wish I had a complete list of ingredients to avoid, but usually I just play it really safe and try all cosmetics before buying.
Fabric softener / dryer sheet fragrance is the worst, for me. All the others are only a problem on myself / my sheets / sometimes on my partner, but dryer sheets make it hard for me to be in the same room as someone who uses them.