I also was given a syringe to clean the sockets, but after several days during follow up/socket cleaning. How about substituting sodium bicarbonate mouthwash with saline rinse ? Not sure if it is correct or not, but my dentist advised against sodium while sockets are healing, and said that salt provides more healthier environment for healing. It must be lukewarm water and not too concentrated, also do not woosh your mouth too hard, so you do not dislodge sockets. I did saline rinse after every single food intake.
I hope it heals soon!