My own sugar cookie recipe is quite dry as well. It cracks along the edges when pressed or rolled and is easily "broken".
I've never experienced a problem with the dough being so dry it doesn't take to cookie cutters, but if your dough is literally falling apart you may want to just spritz (or in the absence of a kitchen water spray bottle, sprinkle with your finger tips) cold water on it. (Like adding water to pie dough) Give it one sprinkle/spritz and kneed it in. Repeat until it just stops breaking. Don't go overboard and make it gooey.
I'd use water over oil or milk because it's the least likely to change the structure of your cookie in the baking process. Such minuscule amounts of water shouldn't yield a detectable change in the end product.
If this is a consistent problem with the recipe, in the future I'd decrease the flour by a tablespoon or two and see if that helps.