RCT canal required for sensitivity?

I have jaw pain, ear pain with swelling underneath for a month.i went to the Dr, ENT to check ear and sinus infection.went to the dentist and he said I have crack on lower first molar and told me to extract it.i went to Endo for second opinion.the Endodontist said that she don't see any fracture in my tooth. She is not sure why I have the swelling in the jaw. She did cold spray test. That was sore for 5-10 seconds in that tooth. So she said I have to put crown in that tooth for minor crack. She tested bite and said might be a minor crack.i asked why I have throbbing pain while lying down.she said I have to check one more time and did cold  spray again and more. and I felt very sore for 10 to 15 seconds. She said this tooth is very sensitive and I need to do RCT. Is RCT necessary for the sensitive tooth ? Or crown is enough for that minor crack which she is suspecting.whether this will cause swelling in the jaw? I have swelling for one month with pain.is RCT really necessary? or crown?