The answer is: the effect of SNRIs (serotonin norepinephrine reuptake inhibitors) on anxiety disorders has nothing to do with norepinephrine. The antidepressants which are effective for anxiety disorders have strong effects on the serotonin neurotransmitter, whereas the dual action of SNRIs on serotonin and norepinephrine appear to be slightly more effective than SSRIs (selective serotonin reuptake inhibitors) for the treatment of depression (Papakostas et al., 2007).
So the effectiveness of SNRIs for anxiety disorders is mostly about its effect on increasing serotonin in the synapse. But it is not merely about increasing neurotransmitters in the synapse…the key is what occurs downstream.
If you look at the neurobiology of anxiety disorders, the fear circuits connect to the amygdala, which is part of the limbic system of the brain which controls emotions. So when you are stressed and anxious, the amygdala becomes activated and this in turn places the connected fear circuits into overdrive. These hyperactive neuronal bundles in the fear circuits are connected to the amygdala, which in turn connects with other parts of the brain. An activated amygdala then activates the sympathetic nervous system and the HPA axis (hypothalamic-pituitary-adrenal axis), and this culminates in the fight or flight response. The subsequent release of stress hormones from the adrenal glands then interacts with the brain and fear circuits to further mediate the anxiety response.
With regards to how SNRIs (and SSRIs) work for anxiety disorders, there are serotonergic neurons that project from the raphe nuclei (located in the brainstem) to the amygdala (located in the temporal lobes bilaterally). So these serotonergic neurons, originating from the raphe nuclei and connecting to the amygdala, have an inhibitory effect on the amygdala. So when an SNRI (or SSRI) is ingested, it increases the serotonin in the synapse of these serotonergic neurons originating in the raphe nuclei, and it binds more serotonin on post-synaptic serotonin receptors, which has an inhibitory effect downstream to the amygdala, thereby slowing down an overactive fear circuit.
However, you are right about the assertion that norepinephrine is stimulating. Excessive stimulation of the locus coeruleus by the amygdala will trigger the sympathetic nervous system. The sympathetic nervous system then activates various body organs that it directly innervates, such as the heart, and the sympathetic nervous system activates the heart to beat faster and contract harder. Norepinephrine is released from the sympathetic neurons, and much of the norepinephrine spills over from the synaptic cleft into the bloodstream. So this excessive norepinephrine can lead to anxiety or panic attacks, increased heart rate, tremor, sweating, hyperarousal, and nightmares.
To block the effects of excessive norepinephrine, you can consider taking a beta adrenergic blocker such as propanolol or an alpha adrenergic blocker such as prazosin to block the effects of too much norepinephrine.
But as stated above, the effect of the SNRIs and SSRIs on reducing anxiety is from the serotonergic action of these antidepressants, not the SNRI’s noradrenergic action.
In summary, the SNRIs are effective for anxiety disorders because of their effect on serotonin, not norepinephrine. Serotonergic neurons originating from the raphe nuclei has an inhibitory effect on the amygdala and fear circuits. So an SNRI (or SSRI) will increase serotonergic neurotransmission in the serotonergic neurons orginating from the raphe nuclei and projecting to the amygdala, thereby inhibiting the amygdala and fear circuits, with the final outcome being reduced anxiety symptoms and a diminished fight or flight response.