If you and your pooch are in some serious need of grub, consider hitting up a dog-friendly restaurant or doing a short patio tour. Plenty of places in South Carolina are dog-friendly, but in the rain, you'll just need to find the ones with covered patio areas like the Lost Dog Cafe. It can be a calm and peaceful way to spend a rainy day, as long as it's not too windy, and you and your dog will get to have some delicious food in the process!
For those who aren't all that adventurous or just want something simple and dry to do, check out some local pet boutiques and barkeries/bakeries. Since most of the owners are dog-centric people, it's a great place to get info on new or dog-specific toys, food and tons of other stuff.