If you experience some anxiety symptoms when starting thyroid hormone replacement for hypothyroidism, your case is not unusual, in fact was my experience when being started on thyroid hormone therapy about five years ago. Patients sometimes get hyperthyroid type symptoms as their body adjusts to thyroid medication and can feel worse when anxiety is already there to begin with. Some Doctors help patients through this adjustment period, by giving them an as-needed anti-anxiety medication, for use short-term. I say short-term because these as needed anti-anxiety medications can lead to dependency upon them, if taken for more than a few weeks.
Strangely, hypothyroidism can cause anxiety symptoms and some patients seem to experience anxiety when it is at the level between normal and sub-clinical. The hormone therapy actually takes a patient to that point between euthroid (normal hormone levels) and sub-clinical hypothyroidism before it goes on to correct the hypothyroidism. This is due to the fact that as the thyroid medication is brought into your system, from the outside (orally), your own thyroid begins to shut down any of its own production of thyroid hormone. Some refer to this as suppression of the thyroid gland or in causing it to atrophy (dwindle down). Once a patient gets past that break-even point on thyroid hormone therapy, they can then see improvement in symptoms as the hypothyroidism is corrected from that point forward. It is a strange but interesting phenomenon and is why it is referred to as hormone "replacement".
This is just my theory as a well studied layperson on the thyroid disease subjects but I believe that in-between point of sub-clinical hypothyroidism and that of becoming euthroid, causes adrenal surges, much like people get with hypoglycemia (low blood sugar). The body has an incredible system of sensors, which are the neurotransmitters and the hormones and it knows how to compensate for hormonal changes. My belief is that when the body senses small downward fluctuations in thyroid hormone, especially when a person it teetering between being euthroid and sub-clinically hypothyroid, the body tries to compensate for the hormone fluctuation, by releasing more adrenaline, as an alternative source of energy and to keep the body better kick-started. I also believe this is why hypothyroid patients at this point of sub-clinical hypothyroidism, will feel these adrenaline surges upon waking in the mornings. As soon as the patient is awake and ready to start the day, the body senses the need to compensate for inadequate thyroid hormone metabolism, for getting them through the day, so sends the adrenaline to help out.
This theory of mine comes from corresponding with literally 100s of thyroid patients, in addition to my own experience with this. It also comes from the reading of a number of medical research articles that clearly state that anxiety and anxiety disorders, can be related to hypothyroidism, especially the autoimmune type (Hashimoto's thyroiditis), which is the most common type. Some research also associates anxiety with sub-clinical hypothyroidism.
You may need help in getting through that in-between stage with the thyroid hormone therapy and if needed, take an as-needed anti-anxiety med, to help you get there. Also, by self-educating yourself, you will lend towards better treatment for yourself as well by becoming a partner with your Doctor in working toward getting your hypothyroidism treatment optimized for you as an individual and unique patient.