I will tell you right now what you have. Inflamed bursa. It will come under the umbrella term SIRVA. Although when they see you, they will tell you it's not. Demand an ultrasound. You may have a fight on your hands. But if you can get a scan it Will more them likely show your bursa is inflamed. With your timeline of events. The injection was likely too high up. If they don't scan you demand one, if they still refuse, get a private one done. They may try to fob you off with you have a shoulder issue and try to give you rehab exercises. It will do diddly squat. Months of anti Inflammatory medication is the way to go or cortisone. Although cortisone is hit and miss with it. You will want ultrasound guided cortisone injection. Where they put the needle exactly into the heart of the inflammation.
Did it hurt more than normal when getting the jab?
If it is that, and you get a confirmation, you could look at claiming for clinical negligence.