I would go by how you feel if you have no syptoms - usally i dont feel normal unless im around 1 so when i go to 2 and still within the normal range i dont feel normal and i go by how my symtoms are doing / how i feel if you feel fine then good then you know where your normal is not what the normal limits is.
The currently accepted range for TSH is 0.3-3.0 according to AACE recommendations. Many people still disagree with this range and feel the upper limit ought to be reduced and the lower limit lowered as well.
However, comfortable levels of TSH vary a lot individually. I would consider your TSH flat out normal, unless you are having other hyper symptoms as well as anxiety TSH is a very poor indicator of thyroid dysfunction since it's a pituitary hormone and several steps removed from thyroid function. The only way to test for thyroid dysfunction is to test free T3 and free T4 levels. These are the actual thyroid hormones and will give you a much better thyroid profile. In addition, thyroid antibodies (TPOab, TGab and TSI) can be tested to see if you are in the early stages of an autoimune disease that has yet to affect your thyroid levels (but could nevertheless be causing symptoms).
I would not look at an 0.93 TSH result and say that it is in any way abnormal. "1.5" simply does not work for everyone. However, since your doctor is trying to find the cause of your anxiety, it would probably be a very good idea to have the thyroid panel. It will help her to rule out possibilities. In the meantime, however, rest assured that there is nothing wrong with your TSH level.