Why doesn't my wrist tendonitis seem to get better when I stay off of it?

It depends. It may be that the injury is actually more serious than tendonitis and thus is not responding to rest. It is also possible that even though you are resting it, it is still being used daily in everyday activities that prevent full healing. A temporary brace or cast immobilization may be required. If your pain is persistent or not improving, I recommend seeking orthopedic medical evaluation.
Proper diagnosis?? It is very hard to give you an answer about your original condition based solely upon your description. It sounds like you had an injection for carpal tunnel in your wrist and you are wearing splints for that as well. I typically tell people not to wear splint all day and all night as sometimes when you go to take the splint off, the comlete immobilzation all the time lead to pain when you start to loosen up and the natural reaction of course is to put the splints back on right way...Leading to more time in the splint and the same issue again and again. I often remind them to take the splints off for a few hours at a time depending upon their activity and avoid making the splints too tight at night. A tight splint at night can increase the pressue on the nerves an injection usually helps with in a few days but in severe casres it may take few weeks finally rarely an injection can irritate the nerve. Id talk to the health care provider that gave you the injection and see what your next action should be perhaps you have nerve conduction studies scheduled or a followup visit to ascertain the effects of the injection. Perhaps they will add anti-inflammatories to your regimen to augment the effect of the injection and the splint.