There many different causes of dermatitis. You would need to have the dermatitis diagnosed to determine if it would be a problem for your dog. I would suggest that you keep the new dog isolated away from your own dog for a couple of week while you start therapy.
DERMA-IONX has been helpful for a variety of different skin conditions. DERMA-IONX is an oral homeopathic product which stimulates the enzyme systems in the body that aid the skin and helps with assimilation of nutritional components in diet and supplements. What sets it apart from most homeopathics is that it is in an oligotherapeutic mineral base which has had a charge added to it.
Initially I would give a half dose of DERMA-IONX on the breakfast and supper food. After there are signs of improvement you can give a whole dose once a day.
I would also give PROBIOTICS to aid digestion and the immune system. I would also give OMEGA 3,6,9 to reduce inflammation and aid the skin and coat.
The water you give should be filtered if it contains chlorine. Chlorine is not good for the intestinal flora.
Diet can also influence the skin. Sometimes the flavor even makes a difference. Please read the label to check what proteins are in the food. When there are skin problems I usually choose a grain free food. If the dog is warm and seeks cool places then I would choose cooling proteins such as fish or turkey as the protein source according to Chinese food therapy. One of these might help the skin the skin heal more quickly because the skin is usually warmer than usual and it will benefit from cooling. If the dog is cool and seeks warm places then chicken, lamb, or venison might be better. Beef is neutral. If the dog is allergic to a protein do not use it even if it has the correct thermal properties. Any food changes need to be phased in gradually, starting with a little new food added to the regular food. This might not make a difference since the dog has been in the shelter. Please keep us updated so we can help guide you.