My dog, who is a Schnauzer and Terrier mix (we don't know what kind of terrier), had extremely sensitive skin, to the point that he will yelp and sometimes bite you when you brush him. But his fur is so wiry that unless we brush him it gets really tangled, and he gets stuff in it, like his own poop, and the mud that he digs in. We've taken to just shaving him down, which does make his fur more manageable as well as making him look a bit less like a Schnauzer. I'm wondering if there's anything I can do to make him less sensitive, or, failing that, if there is a type of brush I can use that won't hurt him. Because even shaved, he still needs to be brushed.
If the sensitivity is all over the dog's body, it's more than likely an allergic reaction to the food the dog is consuming. To fix this try to find dry food that doesn't contain poultry(poultry fat is okay) or -if possible- with only lamb as the protein ingredient. Also you can switch the water it drinks, no more tap water.
Also, you can change the water you bathe the dog with to distilled water and hypoallergenic soaps.
If the irritation/sensibility is localized it's probable the dog it's rubbing somewhere that has organisms or components that cause the irritation, pay close attention to your dog and see it's daily routine, you'll be able to find where the dog is scratching or rubbing and then remove said item from the dog's reach, again the distilled water and the hypoallergenic soaps can help while the sensitivity decays.
I must add that cutting your dog's hair with clippers might not be the best idea, as clippers going too close to the skin will only make the sensitivity worse. Try scissors instead, it's more tedious but your dog's skin will thank you.