I know that it can be just as uncomfortable for you as the experience may have been for your patient, but these things happen. Many new patients are nervous about acupuncture initially and those with anxiety disorders are, perhaps, even more apt to find it difficult to relax during the treatment.
If there is a "fault" on the part of the practitioner in these cases it is generally with their needling technique. For someone who is particularly anxious, and just generally during first treatments, it is often beneficial to needle more lightly than you might otherwise. You may also consider skipping or going very easy on the manipulation of the needles. Just get the needles in quickly and efficiently without causing any discomfort or agitation. I will often use thinner Japanese needles on these types of patients initially, and sometimes permanently.
The points that you chose are fine and you should be able to needle them without the person feeling anything more than a mild pinch - if that. If you cannot, use a Japanese or lighter gauge Chinese needle (36,38) with the insertion tube.
Even if your training expects you to use strong needle manipulation and other deeper needling techniques you have to relate these practices to each patient. Once a patient has one or two favorable experiences you can generally do more advanced needling techniques, if necessary, without making them uncomfortable.
As acupuncture is so helpful for anxiety disorders it is important that you work to ease them into the treatments. Over time, as their condition improves and they become more familiar with acupuncture (and relaxation in general) they will derive much benefit from the treatments.