That is sooooo true about rinsing!! You can never rinse to much
It can take four to six weeks to see a change in skin and coat with a diet change. There are tons of supplements you can give dogs to increase the fatty acids in their diet. Be careful about supplementing a puppy though. A client of mine gives her dog 1 spoonful of tuna every other meal and has seen great improvement in the coat. I'm surprised that the breeder wasn't more specific on what your baby was eating.... Thats very odd.
In Florida we see a lot of people who move down here with their dogs who have no known allergies or skin issues only to find that their dog's skin goes crazy shortly after moving here. We have a lot of skin issues in our area that seem to be climate related. The snow birds that vacation here in the winter from up north have dog's that are fine in the summers but when they are down here in the winter... well you get the idea. I would add a good conditioner to your oatmeal shampoo
That might help as well. Also keep in mind that some dog's have skin sensativties to dyes, soaps, and fragrances. So if you haven't already purchased a shampoo try and find a hypoallergince one that is dye and fragrance free just to start off with.... just in case.