d/d is a prescription skin diet, meaning there are NO contaminants that aren't listed on the ingredient list.
When your dog has been on the d/d has he been getting any other treats or chews? (cookies, rawhides, deer antlers, etc?) My guess is yes, since you're making him a beef stew. Why feed him a novel protein/card diet only to add other things? Didn't your vet explain the process of an elimination diet? When starting a new diet as part of an allergy trial, the dog is meant to eat that diet and NOTHING else. So before you start switching his food again, try doing a proper 12 week elimination diet trial.
Hills and Science Diet are owned by the same company. Hills is their prescription line. While many people disagree with their ingredients, you can't argue with the fact that their diets generally accomplish what they set out to do (unlike the 928392 diets out there with falsified or misleading ingredient lists). Not every diet works for every dog, but in case of true food allergies, why risk ingredient contamination?
Also, there's a strong likelihood that your dog also has environment allergies. For most dogs, it's things like grasses, molds, tree pollens, and other organic things that trigger a response. I've had one dog come back with strong allergies to cat dander, and another to wool.
Either way, I would start a TRUE diet trial and find yourself a veterinary dermatologist. Someone who specializes in skin issues. It's a long, frustrating road but the earlier you start with a derm vet, the better off your pup will be.