Wellbutrin is for depression, not anxiety,
and as such can be stimulating. Drugs like Effexor are for both, but all can cause weight gain. You won't know until you try. The least problematic are probably Prozac, and then Lexapro, but even Lexapro has been known to cause significant weight gain and Prozac isn't usually that great for anxiety, but who knows? They're weird, these drugs, and affect everyone differently.