With food made for allergies, the protein is broken down in a way that even if the dog is allergic, it shouldn't affect them. This doesn't always work though, so it's hard to say if she could be reacting to the Science Diet.
She is young for food allergies but I don't think it is impossible.
Because her skin is oily and she is flaky, I think it might be something else.
If your regular vet can't get it under control, I'd ask to be sent to a dermatologist.
Since it has been so long, I would not be happy that the vets haven't done blood work to at least look at the thyroid and maybe the liver. The results will probably be normal but I think it needs to be done.
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