The dog could have a thyroid problem. The only way to find out is a blood test. Could also have an allergy to something outside. Dogs can have Benadryl (same pills as people).
The dog food change is a good one. We use the brand "Taste of the Wild". It's about $42.00 for a 30 lb. bag. Many times it's the grains in the foods that give dogs problems. This food comes in buffalo, Salmon, & a fowl type. It has been great for all 3 of our dogs.
Another thing that we have had GREAT success using is people "BASIS" soap. $1.25. It's a bar soap. It has alot of fatty acids. I make a lather of it after the dog is wet & make sure I rinse with just warm water very well so no residue is left on the skin. I also do their feet only in a bucket sometimes because that's what is the worst for ours.
This soap works great for allergy & allergy/redness rashes on dogs.