Expert opinion - should penicillin antibiotics capsules treat my dry socket?

Penicillin is. An antibiotic commonly used to treat oral infections. A dry socket is basically the loss or breakdown of the blood clot from the socket resulting in exposure of the bone resulting in severe pain. Obviously no alcohol or smoking following extractions is mandatory. Hope this helps.
No. Not really. You may have a secondary infection there, but the primary cause of a dry socket is loss of the initial healing clot. The extraction site needs to be irrigated and medicated to develop a new healing clot and cover over the bare exposed bone.