For infected tooth, is root canal a right solution?

Hi doctor,

I had my lower right side first molar extraction about a month ago. My dentist prescribed a course of antibiotics. The pain persisted after I finished the antibiotics course, so the dentist again prescribed a five-day antibiotic course. A week went by with no pain, and then the pain re-occurred. My medical doctor prescribed the third course of antibiotics now. I fear an infection under the gum line that is not being addressed by the antibiotics. Since I started the latest course of antibiotics, the pain has been on and off. When I have the pain, it bounces around even to the head. I feel there may be some nerve damage when the tooth was pulled. I am taking Clindamycin for this issue. Also, I am on Insulin for type 1 diabetes, Cyclobenzaprine, Levothyroxine, and Lisinopril. My regular dentist wants me to see an endodontist on the theory that one of the neighboring teeth is infected, and their solution is to do a root canal. I hear that the root canals are a bad idea as a dead tooth is left in place to seep bacteria into the body. If possible, Can I get an antibiotic injection at the site of the infection? Would you please give me your thoughts? Thank you.