Your doctors have erroneously put on board the TSH roller coaster. TSH is a pituitary hormone that is affected by so many things that at best it is only an indicator, to be considered along with more important indicators such as symptoms, and also levels of the biologically active thyroid hormones, Free T3 and Free T4.
Your first test results were only for TSH and Free T4. The somewhat high TSH and low in the range Free T4 are indicative of primary hypothyroidism (Hashimoto's Thyroiditis. To check for that two tests are required: Thyroid Peroxidase Antibodies and Thyroglobulin antibodies, shown as TPO ab and TG ab.
Your doctor reacted to only TSH and proceeded to medicate you based on TSH only, judging by the continual changes in dosage. That doesn't work. A good thyroid doctor will treat a hypo patient clinically by testing and adjusting Free T4 and Free T3 as necessary to relieve symptoms, without being constrained by resultant TSH levels. Symptom relief should be all important, not just tests results, and especially not TSH results. Many of have found that symptom relief required Free T4 at the middle of the range at minimum, and Free T3 in the upper third of the range, as necessary to relieve symptoms. Your current FT4 and FT3 levels are much too low in their ranges.
You can read about all this and become your own best advocate by reading at least the first two pages of the following link. I also suggest reading further, to go through the discussion and the scientific evidence supporting the suggestions on page 2. You can also consider giving a copy of the paper to your doctor and requesting to be treated clinically. If the doctor disagrees after reading the paper, then you will just have to find a good thyroid doctor that will treat clinically as described.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Last for now, hypo patients are also frequently deficient in Vitamin D, B12 and ferritin. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70. You should test for those and supplement as needed to optimize.