Is a vet an option? They will be able to give you a far better diagnosis than we can. I'm getting concerned at this point that her infection is pretty far advanced.
Cankers are the sores. I'm not sure about a parasite that would be large enough to cause a lump in her neck. That doesn't sound right. Can you post a picture?
She definitely shouldn't share food or water with your healthy birds. I have to admit, I don't have a lot of experience with diseases, so you want to read the instructions on the medication. I do know that quail heal quickly, so I would expect to see a difference within a few days.
@007Sean...
@007Sean - Any idea what the spots in her mouth are?
That feed is not sufficient for quail. It's only 16% protein and made for chickens. Can you get a turkey feed for her?
Some white in the poop is normal.
If you're feeding a good, balanced feed, they shouldn't need anything else, so more than 10% of other food can throw off the balance and cause malnutrition.
Do you happen to have the nutrition information for her feed? I would stop feeding her the egg, but the eggshell is okay for extra calcium. One full egg for three birds is going over the no more than 10% treats rule for healthy birds.
Do you have pictures of the white spots?
It sounds like...
How old is she?
What are you feeding her?
What does her poop look like?
How many quail do you have and are any others showing any symptoms?
How much space do they have?
If she is not pooping, has she laid an egg recently?