I'd be going to an emergency vet asap. Fly traps have all sorts of chemicals in them in addition to the stickiness it's self effecting them negatively by irritating their skin and possibly even causing tears in the skin from the trap ripping it.
Also as far as I know Dawn dish soap is the only soap that's proven to be safe on birds,so that's also extremely concerning that that's now on their skin as well. Please get to a vet as soon as you can for supportive care. They can crash and go downhill so quickly from incidents like this. Please keep us updated and ill be thinking good thoughts for your little one.