My two German Shepherds are 5 years old and have had this "paw chewing" habit for the past 2 years, along with recurring ear infections. However, the paw chewing and ear infections didn't occur at the same time. The ear infections were treated with aniti-biotics and one dog had to have her right ear packed. BUT...our vet couldn't figure out why both dogs were chewing their feet all the time to the point the skin between the pads became reddened. He suggested we go to the drug store and purchase "Nizoral".
Nizoral is a "shampoo" that kills the fungus that can cause dandruff and seborrhea.
Our vet told us to wet the paw(s), rub on some of the shampoo, massage it so it goes in between the pads and leave on for 10 minutes, but don't let them lick it. Then RINSE THOROUGHLY to remove the shampoo. We were to do this once every 3rd day and it worked! We only had to do it TWICE.
Also, the vet suggested we try another food and recommended "Wellness" brand, specifically the 'fish & sweet potato' flavour. Because my dogs were having recurring ear infections and the itchy feet, he suggested the food. I was told that some dogs can't tolerate beef and alot of other common ingredients found in the majority of dog foods and could be causing the 'itchy' foot reaction. He said the new food would take about 3 months for us to notice a difference. We changed their food to this Wellness brand and specifically the 'fish & sweet potato' flavour and they haven't had a single ear infection since. We also keep a bottle of Nizoral on hand now for future use.
Also...an elderly neighbour has a black poodle who was also chewing his feet incessantly. Her vet did not know what to suggest. I told her about the Nizoral, she tried it and "Louis" the poodle hasn't chewed his feet since and her dog has been paw chewing for 8 years!!
Hope this is helpful information for you. If you decide to try the Nizoral, I sincerely hope it works as well for you.