Hi Phil,
It sounds like she gets into mischief! Cockatoos are what is known as a powder bird, which means they have special powder producing feathers that coat their main feathers with a protective coating. This means they should not be soaked down on a regular basis. In a pinch, I would just wash the dirty area with original Dawn dishwashing liquid, and then rinse it well. This is what is used by wildlife rehabilitators after oil spills. There are shampoos available online that are marketed for parrots. But to be honest, I see SO many dangerous products for parrots online, as well as terrible advice from bird owners. I personally would stick to Dawn as that had been safely used on wildlife for decades. Again, just use the original basic Dawn – if unsure, contact the company as they do promote it for wildlife and donate to wildlife causes.
I hope your seagull has found a flock to live with!
Thank you for asking Lafeber,
Brenda