My personal opinion: Don't use Dawn!
Here's an example of one of their products with toxic chemicals.
Dawn is also owned by Procter and Gamble, a company that is KNOWN for testing on animals.
I make all natural, organic and sustainable soap with organic food grade ingredients (I eat my own organic coconut oil which also I make soap with, lol) and I don't use my own soap when bathing my parrot, lol. You "could" put organic aloe vera juice mixed with water in a bottle and spray your bird. Or, take your bird into the shower and see if she wants to bathe with you. If that fails, offer a water bowl to bathe in, or again, use a spray bottle.
I once rescued a pigeon that was all white with a black tail. He was filthy with a wounded wing and couldn't fly. Over time, as I nursed him back to health, he molted and his plumage was bright white. He only bathed in water, once.
Here is a photo of him, healthier:
