If the steroids help this could be in part due to allergies and if the antibiotics help it could be in part due to infection.
Paw licking and skin chewing is often a sign 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, 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.
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
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. These can help with the odor by eliminating the infection on the skin.
Dogs can also have low thyroid level issues and that can trigger skin allergies. The vet can check for that with a blood test. This will turn the skin color dark in many dogs. You can read about low thyroid here
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. Frontline could resolve a lot of the reactions for you.
Hope this helps you!