What you describe could be allergies but it might also be something such as mange mites which you can read about here
Or a skin infection with ringworm
or a bacterial skin infection possibly due to a low thyroid level
Paw licking, itching and skin chewing are often signs of allergies. You may see red staining to the fur in the licked areas or raw spots where the dog chews itself.
The skin itches from allergies, the dog licks and chews trying to soothe the itch, making the skin raw and allowing other infections to set in which may itch even more.
You can read about allergies and dermatitis in dogs here
Dogs can develop allergies to foods, even ones they eat all the time, and to inhaled items, and contact allergens such as rug cleaners, cedar beds, or chemicals including lawn chemicals or even flea bites.
You might want to try a different dog food that has no ingredients the same as what you feed now. Diets of Fish and Potato, venison, or rabbit etc. where the protein source is new and there are no grains in the food can work for many dogs for example. Another option is the hypoallergenic diet from your vet Hills ZD. Changing food does no good if you feed the same ingredients. Remember food changes have to include all treats and can take 8-12 weeks or more before you see results in skin and coat. You might stop the supplements as if a dog is on a good food they don't need them.
If this is an inhalant allergy you may find using a HEPA air filter in the room the dog uses most and wiping the dog down with a damp towel when it comes in helps reduce allergen exposure.
You might want to see if some plain Benadryl helps with the itching. A common low dose is 1mg per pound of dog every 12 hours. Do check with your vet about using a med but this one or another may help him a lot. If you choose to use that please read here about cautions
I know you said you don't want to use pills but itching is pretty awful to live with.
You may want to consult with your vet and consider doing allergy testing.
You can try a Chlorhexiderm shampoo from the pet supply store in case the problem is a bacterial skin infection. Malaseb is another good one to eliminate yeast from the skin. Follow directions on the bottles if you try one.
Dogs can have low thyroid level issues and that can trigger skin allergies and infections. The vet can check for that with a blood test.
If your dog is not on a flea prevention that might also help. Just one flea bite can make an allergic dog itch all over. If that is the problem then Frontline could resolve a lot of the reactions for you.
Some people find their dogs do better when they are given omega 3&6 fatty acid supplements such as Derm Caps or Linatone from the pet supply store.
If your regular vet is not able to help you then you may want to see a veterinary dermatologist and this page may help you locate one if your vet can't refer you to one
Hope this helps you!