assuming that you have tried a lamb and rice formula of one of the major brands...then I would try Loyall made by Nutrena..One of my girl shorthairs had this same sort of problem...she had the loose stools, threw up after every meal, but was always hungry..she even would get scabs on her skin. My dogs vet said it must be an alergic reaction and that I should try switching dog food brands I tried seemingly every brand of dog food...not the cheap ones either and nothing worked. I finally asked my horse vet what she would recommend...She said that most dog food is processed and bagged by the manufacturer with the cheapest bid made to the producer...In other words a dog food may be bagged in a dog chow bag(just an example...all brands do it) but produced anywhere and possibly more than one plant...even over seas...ie the china dog food tainting...She told me to get a dog food that was made in the country at the same plant every time...this leads to a more consistent diet...Sure enough I switched to a River Run brand made by Nutrena....in minnesota,I think, Problem over...2 months later you wouldn't be able to tell that she had been "sick"..The dogs vet was amazed that she could turn around like this while eating a "cheap" dog food...
One more thing now that I think of it...I remember the dog vet saying that the ration that I switched to was lower in protein than the "good" bagged food....little did she know, once a day I give a raw egg to each of my dogs on their dry dog food....this raises the protein level and increases palatabilty...confirmed by my horse vet. I know my eggs are hormone/drug free...while instead "dog food producers" use horrible things to raise the protien levels in their rations....the key word left out horrible and cheap.
moral of the story buy feed from a feed company...not a major discount store chain.