Any doctor that would ignore your symptoms and not even run tests is a waste of time. So yes you should move on to another doctor. I highly recommend that you read at least the first two pages of the following link, and more if you want to get into the discussion and scientific evidence supporting all that is recommended. That will give you info on the tests that should be done and what to expect from a doctor.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Also, a good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T3 as necessary to relieve symptoms, without being influenced by resultant TSH levels. You need a doctor that will treat clinically, as described. If you would give us your location, perhaps we can give you the name of a doctor that has been recommended by other thyroid patients.
We know of doctors in some areas of South Carolina. Where are you located?
Well, being in the UK puts a different slant on it. Doctors there are under the dictates of the NHS which basically depends on only TSH for diagnosis and even treatment. That does not work, of course, for many people. One of our UK members some time ago told us that the only way she was finally successful in getting treatment was to do enough reading to be able to tell the doctor what you know and what your symptoms are and what you need in the way of treatment. She said if you do that and push hard enough that you can finally succeed. So you can start by reading the link above and asking any questions you have. We will be glad to help any way we can. If you are totally unsuccessful with doctors, then you might have to go private. If that is necessary, the only good thyroid doctors Inow are in the London area, if interested.
Please post results and reference ranges for any thyroid related tests done for you.