My older dogs paws are puffy & he always licks them. FOOD allergies or other

[Alexandria]

type of allergies? Other allergies like grass, dust etc. and is the only way to find out by doing expensive allergy testing? Everytime I take him to the vet he gives me antibiotics and a topical spray. He also has a skin problem. I am now trying a different expensive dry dog food only FISH with NO GRAIN to see if that will help him at all.
95 lb. pit bull mix.