BudgieBill... Welcome. This is a really great site, I'm new to budgies & reading all of the "stickies" in each section was a huge help to me. Hope your girl feels better soon. I went through mite treatment with my budgie, and, he is white and light blue and was a bit dirty in the beginning of owning him. He didn't like to bathe much (still doesn't) but I offer various bath dishes on a regular schedule because once in awhile he likes it and wants to... But not much, maybe once a month or even less. I've read that some budgies (and a lot it seems) don't like to take a bird bath. Some do, some don't apparently.
I overlooked his dirty head and dirty wing feathers and he cleaned himself up a lot, and also over time new feathers came in and his messiness was a thing of the past.
Also, my budgie eats a lot of greens (which is great, yay!) so half the time his beak is stained green and sometimes his white feathers around his face have a bit of a green tinge. But this is actually a really wonderful thing. Some budgies won't eat vegetables and that's not good for them, so at times a messy budgie is a healthy budgie. (Sometimes they might have an orange face if eating carrots.
At first I was concerned about his appearance, but then I realized as a new owner there's so much else to focus on in the beginning, things that are so much more important...healthy diet as Aluz mentioned, safe cage & toys, making sure the budgie feels comfortable & safe etc. ... developing a slow, safe & secure relationship etc.
I focused on the other things and the dirty feathers took care of themselves.
Congrats on your new budgie... she sounds like she found a good & caring home.