Hello Megan, I would be happy to help you troubleshoot your grandmother's butter cookies. When cookies spread too much and turn flat, it is almost always caused by a few common temperature or measurement issues.
First, the butter might have been too warm or completely melted when you mixed the dough. If butter is too soft, it melts too quickly in the oven before the flour structure can set. Second, your baking sheet might have been warm from a previous batch, which starts the melting process early.
To fix your remaining dough, try chilling it in the refrigerator for at least 30 to 45 minutes before baking your next batch. You can also mix in just a couple of tablespoons of flour if the dough feels overly greasy or wet, but chilling is usually the safest and most effective method to save your ingredients without wasting anything.