Anxiety, in all it's forms, is far too often interpreted as an indication of weakness, dysfunction, and hyper-sensitivity. But while anxiety certainly doesn't feel good, it is essential to a healthy life and often an byproduct of empathy, compassion, conscience, and insight. It's also the built-in alarm system that has kept us alive and thriving since, well, forever. The challenge is not to eliminate anxiety from your life (see Q/A above), but to embrace and harness anxiety so that it becomes your trusted internal ally and a self-regenerating source of motivation.
Truth is, no one feels perpetually happy, wonderful, and optimistic. (Well, nobody that's not deeply immersed in denial or void of self-awareness.) Ironically, in my experience, it seems that it's often the more intelligent, emotionally fluid people that wrestle with worry, stress, and anxiety the most.
Empathy, compassion, insight, and even creativity are indeed double-edged swords. The same qualitites that enrich your life also leave you absorbing the angst of the world around you. And unfortunately, you can't have one without the other. Sorry. Conversely, sociopaths and narcissists -- who lack empathy and self-awareness -- rarely enter therapy and if they do, it's typically either involuntarily or because they're searching for a way to manipulate someone.