There are a lot of tools that you can use for shading or blending charcoal on paper. A blending stump (paper stump) and a tortillion are the most popular and the easiest ones. They are almost the same, except a blending stump is a stick of tightly rolled up felt paper with two pointed ends, while a tortillion is made of stiff paper and only have one pointed end. The main difference between them in the process of drawing is that a stump does the shading softer, due to its soft material, but a tortillion can do more directed shading lines. After a while these tools can become dull and dirty, and artists make a big mistake by throwing them away. If you want to know why it's a mistake I am ready to tell you My Tip #6.
As you can see from the pictures these tools lost their forms, the ends became dark and dirty. But thanks to that they are priceless for drawing a charcoal portrait. I recommend to keep old blending stumps of different sizes and only one or two tortillions. Since the ends of stumps are made of soft paper, they easily absorbs charcoal. This way they are so good for smudging and smearing even without actually using charcoal. A blending stump becomes a charcoal pencil. You can do dots, lines or light shading spots. Especially when you are drawing a portrait it is perfect for drawing wrinkles or freckles.
I hope you stop getting rid of your old drawing tools and start using them as a beneficial method of shading.