Giving medications for such a long time tends to change the composition of the intestinal flora. It might be worth trying a gentler approach to see if it can get things back on track. The homeopathic DERMA-IONX helps to stimulate the skin to heal. PROBIOTICS help the immune system and tends to crowd out bad flora. OMEGA 3,6,9 helps to reduce inflammation and helps improve skin.
I think it would help to feed a grain free food instead of your current dog food. Purina Beyond makes grain free food (including fish) and can be purchased at many large stores such as Walmart and Target. In Chinese food therapy if the skin or joints are hot, one would want to feed a cooling diet such as fish or turkey based dog food. Only use these if there are no known allergens in the food. One would avoid chicken or lamb which are warming.
If your water contains chlorine you need to filter it. Bathing and drinking puts the chlorine inside the body. Chlorine kills bacteria and allows yeast to grow. These are important influences.