We need to look at the diet first. Second- wet weather can have the grass etc stick to the paws.
First Purina dog chow has more rendered by products and poor grain quality. It sustains life, but so does a leather shoe soaked in vegetable oil. I would put on a lamb and rice diet first- Any of the better brands would be better at this point. It looks like more money- but its not- you are feeding literally half b ecause a better food is more digestable.
Second- if this is a contact allergy from the weather, with a better immune system, the dog will be less likely to react. however- with a really bad case I had many years back, I got a bottle of puppy shampoo ( very mild) and rinse off the dogs feet adding a tiny dab of puppy shampoo to get off the dirt if its really bad. Remember shampoo does dry out the skin so use the rinse more and dry with a towel.
Human corizone has too much alcohol in it for a dog. It will actually dry out the oil in the skin faster than if you did nothing. Really bad cases, the internal veternary medications are better. Better to give at night so the antihistimine is in the body longer with out passing through the urine.
I still would however- first and foremost= take the dog to the vet. If the dog has infected skin, there is no way we could not know that without seeing the dog, and we are not vets.
Foods- Eagle holistic or lamb and rice, or even Nutro lamb and rice would be better than Purina.