Is it possible to get a dry socket after they sewed the extractions sites shut?

Possible, but not as. Likely since the protective blood clot is covered with mucosa. If the sutures come loose & one chews or rinses too vigorously, the clot may be dislodged & food & bacteria can enter & infect the bone. Follow oral surgeon's advice for post-op care. Return if pain worsens.
Yes. A dry socket will present as intense/increasing pain typically 48 hours after a tooth extraction. It is caused by exposed bone in an area that has lost it's blood clot. The suture is designe to help keep the clot in place. But, sutures can fail. Treatment is to have a medicated dressing placed into the affected area. Antibiotics will not help a dry socket.