In early November, I dug my sweet potatoes from the ground, washed and dried them, then stored them in boxes on shelves in an unused bedroom in my half-basement. These (pic below) are the smaller ones that are left after eating them for three months. They are in exactly the same condition as when I put them away.
A few I stored in a basket in the kitchen. These are in similar shape with the exception of two small ones that have developed little roots. I can only imagine it is in response to the ambient light and the additional heat they receive in my kitchen. I chose these two to try to grow for replanting. They have been duly stuck with toothpicks and suspended in jars of water. I'm curious to see what happens with them.
As for the larger sweet potatoes, I continue to enjoy them in various recipes like Spicy Sweet Potato Fries. I'm still tweaking a little, but it is basically cutting the potato into fry shapes, coating in oil and spices, then baking at 450ºF for 12 minutes on a cookie sheet. Salt and pepper the raw fries in a mixing bowl. Stir up a dash of cayenne pepper in either olive oil or coconut oil, then coat the fries with the spicy oil. Spread the fries out on the baking sheet so they don't touch, and keep an eye on them as they come close to the 12 minutes end-cooking time. Ovens vary, so time and temp may have to be worked out. By the way, as long as they are clean, there's no reason to peel them. They are even healthier with the peel intact, and it's not noticeable at all once they are baked.