Do I really need a root canal?

After my dentist did a normal procedure to remove a small cavity, she drilled it and then put the filling in. A few hours later after the anesthesia wore out I started to feel pain, sensitivity to hot and cold especially. So a 1-2 months later I went to a different dentist to see the cause of my pain and he showed me the x-rays of the molar which shows the filling and what seems to be a nerve touching the filling but the image is not clear nor shows the contact. In the end my new dentist told me that my old dentist drilled to deep and caused nerve damage and I need a root canal to KILL my tooth. Do I really need it?