Concerned I need a root canal after recent crown. Is it normal to continue to have discomfort and sensitivity after a crown?

Pain after placing a crown isn't completely abnormal. There are many reasons for pain after crown placement, even if you had minimal discomfort before it was done. One cause is simply the trauma of having the tooth worked on. Another reason for pain might be that the crown is too high and is hitting the opposing tooth, causing it to become sore. If the pain continues after your doctor has evaluated you, it might be time to consult with an endodontist about getting a root canal. By placing your crown on with temporary cement your dentist made it easier for the endodontist to perform the root canal if necessary.

Yes, your symptoms can be very normal following a crown.  Many things can contribute to this.  It sounds like you tooth may have been cracked before the crown was done.  It is very common to have some sensitivity following crown placement.  It can take some time for this to settle down but they usually do.  If you grind your teeth this can take longer.  Patience, unfortunately, is your ally.  If the tooth gets worse you may need a root canal treatment but for now it sounds like normal post op sensitivity.  Good luck!