I love a breed that is so prone to skin allergies. Right now dealing with fleas (while current on Advantix
). I had a dog who suffered with some food allergies as well as flea bite dermatitis. Since your dog is chewing paws and rubbing ears, it sounds more like food or environmental allergies. A move of 100 miles isn't that significant for the environmental changes. My itchy dog traveled about 200 and got her flea infestation.
It can be so difficult to eliminate the food allergens. It is trial and error and can take time. Corn can be a big culprit. I found that one of my dog's triggers was artificial dyes found in treats. Remember to check ingredients on the treat bags. Also, some dogs can be allergic to grass and sometimes this problem flares at different times of the year. This could explain the paw chewing and face rubbing.
A few things that helped us get through the itchies were adding an oil supplement to our dog's food (fish oil can work, but get your vet's recommendation and dose), running a humidifier in the room where our dog slept to add moisture to the air and help with dry skin, and using oatmeal shampoo (we use Hylyte).
Good luck with your dog. Your vet can prescribe allergy medication to help with the itching, but it won't resolve or identify the trigger(s). Please try to avoid using steroid shots as these can have some very negative side effects in both the short- and long-term. Also, some yeast infections can cause chewing and scratching as well, so this is also something you may want to consider. I think this isn't a high possibility. When my dog got a yeast infection it had a very distinct odor.