First of all I would like to make it clear here that you have an injury. It is not something that has developed over years. Definately, if you do not have a fracture
or a dislocation it seems to be more or less of a soft tissue injury
So, therefore there is no need of further x-rays if the first one does not show you any fracture until and unless the x-ray was of a lower quality with bones not visible clearly.
I cannot comment on the hurry to get the CT scan done without being referred to a orthopaedic or sports medicine rehabilitation
department for the rehabilitation of the injury in combination with the medication.
I would suggest you also see Physical Therapist of the conserned speciality to undergo the therapy for the same. Which would be decided after thoroughly evaluating the injury site.
Hoping for a fast recovery.
Orthopaedic & Neurological
Lifestyle Medicine Counsellor