Does my dog have a serious skin condition?

It sounds more like a skin disease than allergies. The dog needs to be examined by a vet. Usually, you get a medicated shampoo that has medicine in it to cure that specific skin disease. I can't guess what your dog has because so many skin diseases look alike but a vet will know. Since the "soothing" shampoo you are using isn't helping, it is time to go back to the vet. By NOT taking the dog to the vet, your mother is setting a bad and irresponsible example for you.