The problem with medicated shampoos is that they were originally meant for horses. You can't take a horse to the vet, the vet has to come to you. This is expensive, so they made an industrial strength shampoo with an industrial strength antibacterial. The idea was that it would take care of minor skin infections. When they marketed the shampoo for dogs, they kept the exact same formula and strength. These shampoos are pretty good at sterilizing the area, but they are pretty brutal on the skin of a Chihuahua or Maltese, or Dalmatian, or English bulldog.
All but a very few are designed for other conditions that don't apply to dogs that lick their paws. Shampoo prescribed for seborrheic dermatitis is dandruff shampoo. These shampoos are in the T-Gel, Selsun Blue family. They are not gentle shampoos. They are made to dry out excess oil and dissolve fat deposits on the scalp. They are especially harsh on dogs whose skin is already dry and itchy. Very few dogs ever get dandruff. The ones that do don't lick their paws as a result.
Only a few are for allergic skin inflammation. One variety contains hydrocortisone. It works pretty well for serious skin itching. It's also safe because all the hydrocortisone goes down the drain and your dog can't ingest it. But it doesn't last more than a few days and the baths require 20 minutes of soaking. It's a good backup plan for bad flareups but isn't really a good management tool. It's more of a bad case scenario.
If you are currently using a medicated shampoo, you can give it a test by lathering up both your arms past your elbows, then rinsing and drying. If your skin feels dry, or uncomfortable, or itchy, it’s probably making your dog's skin feel the same way.
The best shampoo is no shampoo. If you want to help your dog feel better, give him a bath in some lukewarm water with a few tablespoons of baking soda dissolved in it. The baking soda will raise the pH level of his skin. It can often relieve itching for a few hours and calm down irritated skin.
Try to bathe your dog as little as possible. Even the mildest shampoo can wash away natural skin oils. We just mean keep it under once a week for little dogs and once a month for big ones.
If they are really stinky, that's a different story. If there's no visible infection, they may be eating stinky foods. Some game meats and alternative proteins can be pretty stinky. If you currently give your dog fish oil capsules, you might want to reconsider. They probably aren't helping his paw licking or his smell.