When it comes to something that could lead to long-term difficulties - see your doctor. (Yeah, I know you're already going, but I wanted to get my vote in.) Treating yourself may work, or it may only alleviate your current symptoms and allow the problem to get worse.

Forcefully bending the wrist is what causes tendonitis which can lead to carpal tunnel syndrome. In the long run, it would be a good idea to work out a way to play with your wrist as straight as possible, and minimise movement. Try moving the neck and your arm around so you get a better angle, and try to hold the guitar stable with your body and right arm if you can. The pressure on your left thumb can be a problem.

Wrist braces mainly serve to let you know when you're bending your wrist. They won't really prevent you from doing it, and worn all the time can cause their own problems. (Loss of some muscle strength and irritation). Most doctors who are up on current therapies will not recommend they be worn all the time.

As far as cortisone injections - no, it's a Bad Thing to get repeated shots. One shot to nip the inflammation in the bud is another thing. I have a screwed up shoulder (impinged acromium clavicular joint) and went to the doc when the pain first flared up. I got one cortisone shot, which almost immediately alleviated a lot of the pain. Within a couple of days my shoulder was nearly back to normal. I noticed no side effects. The inflammation alone in a joint can cause damage.

There is a bit of information here