Thanks to Emily, I figured out my sugar cookie problem. She suggested that the problem was too much butter in the dough, so when I rolled out the rest of it I was very generous with the flour, and ta-dah! Perfect sugar cookies. The kids are thankful, too. They had gobs of fun decorating the cookies. I think they enjoy decorating them even more than eating them! And, of course, it's much more fun to decorate cookies that have a shape than just big blobs.
Thank you, Emily!