Each person is different. All medications have side effects. You can pay attention to how it affects you and decide with your doctor if the decrease in depression is worth the increase in anxiety. Your doctor can also add another anti-depressant or anti-anxiety medication to corabat the increase in anxiety (if you have an increase in anxiety--you might not). And you're right, you can learn cognitive behavioral things that can help anxiety (and depression) as well.
I also have depression and social anxiety. I had tried Zoloft, Prozac, Pazil, and Effexor. None of them helped me at all. Wellbutrin has been a miracle for me. I still have depression, and I still have social anxiety, but it is manageable. Maybe my anxiety is higher on Wellbutrin, I'm not sure, but it's a small price to pay for relief from my depression. Anxiety is uncomfortable, but depression can be life-threatening.
Talk to your doctor. Don't rule it out based on what you may have heard. Remeraber that the most vocal people are always the few that have had bad results with whatever they've tried. It doesn't mean they're in the majority.