I actually would suggest a trip to the ER (since it's the weekend and you most likely won't be able to get in touch with your doctor) - with teh swelling you're describing, it definitely needs to be checked into to avoid compromised blood flow.
The pain could be from the doc hitting a nerve with the injection, but it also could be at least partially caused by the swelling itself pressing on the nerves.
While it's quite common to have some increased pain following a steroid (cortisone) injection, the pain and swelling you're describing are definitely out of teh realm of what would be considered "normal" for after an injection.
I'd definitely sugggest a trip to the ER - meanwhile - try to keep your hand elevated as much as possible, as that should help some with teh swelling and pain. Don't worry about wearing the splint if it's cutting into your hand - wearing it actually could do more damage if it's that tight. You can try icing down your hand/wrist to see if it helps, but don't use the ice/elevation as a replacement for going to the ER - you still need to have it checked out.
Best of luck!
It sounds like the doctor hit a nerve! Call the doctors office!!