It most likely has nothing to do with the food
A veterinary dermatologist would be your best bet, if that is not within your means, I would work very closely with your primary vet and do exactly as he says, I would disregard any advice that I read online/forums and such, tons of misinformation.
Per the search engine:
/forums/topic/dog-chewing-paws-raw-and-vet-prescribed-food-making-it-worse/#post-116655
/forums/topic/prescription-diet-and-pooping-too-much/#post-109822
PS: There is no cheap way out of this. Veterinary care is expensive nowadays. It is what it is.
From what you have described, it sounds like your dog may need the expertise of a specialist. It may be cost effective in the long run, rather than going back and forth to the regular vet (bandaid treatments) and trying all kinds of different foods with no significant results.
