The reason the packing tastes so bad is because it is basically oil of clove. The dry socket paste is Eugenol. It does taste bad but it usually does the trick. Maybe the dentist could place a piece of gauze in your mouth prior to repacking to help soak up the liquid.
For a dry socket your dentist will clean it and pack it with medicated gauze. He may ask you to see him at regular intervals for addressing change until the socket heals and pain decreases. You may be prescribed antibiotics to prevent socket from becoming infected. You may need a change of dressing. See your doctor as soon as possible. Hope it helped. Keep me posted with more queries.
Take care and regards!