Whether or not they're properly stored you still want to eat them within 1 week of purchasing the artichokes. A great way to preserve them is to cut a bit of the stem off and sprinkle it with water. Sort of the same way you prepare flowers for a vase. Then, store it in an airtight container or ziplock bag until you're ready to use it. Don't wash them beforehand, otherwise, you'll trap too much moisture in the container.