Depends how deep the cavity is. You can always give drilling a try, and then get numbed if you can't stand it.
I had something very similar recently. I had a molar that became very sensitive to cold. I thought I must have cracked it and went to the dentist. He told me it was just "root sensitivity" and it might take up to a year to go away.
It got worse and worse. Pretty soon heat, cold and pressure were all very painful. It got to the point where even pushing on it with my tongue hurt and it was keeping me awake at night. I was traveling, and couldn't make an appointment. Right about at the point where I couldn't stand it any longer and was thinking of making an emergency appointment with some other dentist, it suddenly stopped. I'm kind of thinking I must have had an infection under the tooth or something.