Everyone takes pain meds after a bad extraction like that. You'll have this pain for several more days. The pain meds help it heal by making the area less inflammed. The swelling keeps the blood from going to the area around the tooth to supply healing nutrients to the area.
You need to take some pain meds to help it heal. Ice will help too. It reduces the inflammation too. It can damage the area if you leave it on your cheek too long though. They moved nerves and tissue that is not meant to ever see the light of day and it takes time for it to lose that sensitivity.