If you don't rinse all the soap out, the residue can cause itching.

Also can be allergic to something in the soap?
Is your dog a special breed? Some dogs cannot tolerate gluten
and/or corn in their food. There are some grain free foods available now.
Blue Wilderness by Blue Buffalo is one. We are using the dry kibble for our cats now. Cats have more problems with grain than dogs however.
example

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I know a person who was told by the breeder to not feed gluten to the dog. But I don't know if that is breed specific or not.

There are also mites that infect dogs.

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