Re: Tooth Sensitivity
Which teeth are you talking about? If they both fractured, it is most likely due to severe decay, which may have been present for years. One can have severe, deep decay without pain, but it will eventually catch up and from that point on, cannot be ignored.
Pulpitis (inflammation of the pulp tissue inside the tooth), it if has gone to the state of an infection, can often render local anesthesia impossible until the infection is cleared up. The only way to do this is to take a full course of antibiotics, and then do the treatment, whether it be endo and a crown or an extraction. If your teeth are beyond saving, then it sounds like the fractures may be quite severe, leaving little tooth structure to work with.
The standard practice when a patient is in pain and cannot be treated, is to put them on antibiotics and pain meds until the inflammation/infection subsides and then to treat the tooth. There is no reason for you to feel pain when being treated, but these things CANNOT be postponed just because your pain may come and go. It will not resolve or improve on its own, once reaching the state of disease implied by your description, if I am reading your post correctly. Once the pulp chamber has become exposed, its integrity is forever compromised by the presence of bacteria and your options are either a root canal (to remove the diseased pulp and sterilize the chamber of the tooth) or to extract it.
I also understand and sympathize with those that have dental anxiety, so find out if your dentist has Nitrous Oxide available ("Laughing Gas") for use during treatment. This would be an option to consider and is one of the safest methods of anxiety control available. [It also makes the dentist's job easier and safer to have a relaxed patient.]
Lastly, if this is a financial hardship, consider having the treatment done at a dental school. The schools treat patients for a fee (adjusted according to your financial situation), and treatment takes longer than when done commercially, but the care provided is excellent and involves students and instructors.
I am sorry that this has happened, but please don't ignore it any longer, as it will only worsen, and the infection can spread throughout your body (through your bloodstream) and have serious complications for your heart and other internal organs.
Best of luck.