If the itching is from dry skin, you can use coconut oil, but I use two shampoos from Isle of Dogs that are wonderful. My little girl doesn't have dry skin, but she was biting one foot, and my vet thought it could be allergies, since that is a classic sign. The next vet visit my vet found an injury on her paw, one that my husband or I or even my vet couldn't see initially. We all closely examined that paw. I was convinced it wasn't an allergy, and I was right. I was using Isle of Dogs #10 shampoo or IV SanBernard Black Passion, both very moisturizing shampoos. When this happened, I decided to try Isle of Dogs #12 shampoo. The evening primrose oil is wonderful on the skin and coat. I buy the larger sizes to make them more cost effective. I gave some to a friend of mine whose Bichon was suffering from hot spots, and it worked great.
You should try putting a cup of water close to your heat source. My last Yorkie was really affected by the heat in winter, and my vet suggested I do that. A humidifier would also work. It could be as simple as not completely washing out all of the shampoo.
https://iodogs.com/products/no-12-tr...se-oil-shampoo