I have done karate, though not specifically for anxiety. It was certainly quite exhausting and cardio intensive, so if you think something like that helps (similar to jogging or biking), then it might be worth a try. It's also great to just be able to hit things...
But as for being on display, in my karate class the trainer was almost always showing techniques together with a random student, asking students questions and so on. I'm not sure whether that was his style or is like that in general with karate (he was trained by Japanese karate pros, so I kind of assume it is common).