Most Dentists will only remove the Tooth as a last resort. What did the Xray show?
I had a similar problem, had a Root Canal, but had very painful sensitivity whenever I drank anything. I returned to see the Dentist, he Xrayed me said there was no difference.
I decided to see another Dentist who again took Xrays and found the previous one hadn't filled all the canals and also hadn't gone deep enough. She redid the Root Canal and £280 later I've had no problems since.
She said the sensitivity may have been because the Tooth has a Hairline crack.