I would NOT use vinegar on his skin. It will dry his skin out more than it already is. I would instead go to the grocery and get some oil for his skin. You can use olive oil if you want. Grapeseed is also EXCELLENT and macadamia nut is truly wonderful if he doesn't have a problem with nuts. Or you can go online and get some shea butter.
I suggest the grocery because he WILL lick the spots and you need something food grade. I don't know if dogs react to macadamia nuts, but I know the others, or even rice bran oil (I got a big bottle for $6 at Big Lots) would not harm him. You need to put oil on the spots to help with the itching. Especially as all that washing is totally stripping his natural oils from his skin and coat. Just rub some of the oil onto the spots several times a day.
Rice bran oil is incredible for skin and hair. I have psoriasis and my idiot body reacts to most lotions and creams. I can tolerate a few, like antibiotic ointment or benadryl ointment, but that is about it. My in-laws have a dog that husband refers to as a 'Lhasa shih-t" who has super sensitive skin. They thought I was bonkers sending both a little bottle of rice bran oil and a container of cream I make with 2 parts of shea butter (Refined because the unrefined stuff STINKS, I don't care what they say about it!) and 1 part rice bran oil for the dog and stepMIL both to try. But now all of their friends want my cream and so does their vet. It may seem strange to make cream for your dog, but it will do amazing things for you too. You can use the rice bran oil right on your skin and it is amazing.
If the medicated baths are not working, try putting some old fashioned, unflavored oatmeal (or the quick cooking kind in the canister, just not the packets even if they say unflavored on them) in the bath. You can take an old sock and fill it iwth 1/4-1/2 cup of oats and put it into the water, let it sit and squish it a bit, or put it into the blender/food processor and grind it fine and put it into the water loose. It is amazing for soothing skin and sooooo much cheaper than the packets of oatmeal bath like Aveeno.
Vinegar is only going to dry out the skin more. The drier the skin, the more little cracks in the skin will let in fungi and bacteria, making the problem worse. You need to add moisture, which means water and oil to help seal in the water. So after a bath or a wipe down with water, put some oil on the spots.
I know people are all big fans of coconut oil now, and it has uses, but I would NOT use it for this. It actually will strip the natural oils from the dog's skin (and from people skin) and is much more harsh than helpful. Olive, grapeseed, rice bran, mac nut, almond, apricot, even canola oil is better than coconut for skin issues.
Another thing that may help is to do a final rinse of the dog with filtered water. If you have hard water, it can make any issues with dry skin a LOT worse. You can buy water for this, or just use a Brita type filter pitcher, but just rinsing the dog with filtered water after a bath could be a very useful step. The bath can be in regular tap water, you just don't want the minerals sitting on the skin drying it out, hence the final rinse.