Try to use your left hand more often, not just when you have to write. It'll help build the muscles and coordination up in that hand so it'll be more like your right one. Besides, if your right hand is seriously hurt, this would be the best way to go anyway.
Writing with it does take a lot of practice and a bit more thought that with your right hand, so don't rush into writing something like an essay in under and hour. Think back to how you learned to write in grade school and try that. They did it slow and carefully back then for a reason. Also, don't be bothered if you develop a writing style with your left hand that's different than the one for your right hand...just think of them as different fonts.
Right now, I suggest you switch your mouse from right-handed to left-handed. And I really mean right now. Best time to practice this sort of thing is right away. Mind you, if it does start to hurt or get tired, take a break and maybe see if there's any way you can make it more comfortable.
Yes, I'm still alive...still alive...