I won't quote the preferred labs but will instead refer you to Pinterest! There's a whole lotta info with links to other sites and forums where you will find all that stuff. And there ARE labs you need beyond just T4 and T3! Get the info, print out the labs recommended and go back armed with your newfound knowledge! Best of luck!
PS I just ordered a book specifically for Hashimotos Thyroiditis online... it would probably be a good investment to get some literature too, no just online stuff.
Assuming that the lab's reference range for Free T4 is like many I have seen, your TSH and Free T4 results do not indicate hyperthyroidism. It would have been a good idea to also test for the biologically active thyroid hormone Free T3. It is important to know that also.
If your doctor calls and says everything is normal, you should ask why you are having palpitations, anxiety and insomnia. You should suggest that you need more tests done, specifically at least Free T3 and a diurnal saliva cortisol panel. And I am sure there are others that the doctor will know should be tested, if you push her for an answer to why you have those symptoms.