The likely reason your FT4 did not increase is that when you take thyroid med, the TSH will drop in response. This reduces the stimulation of your thyroid gland and the output of natural thyroid hormone is reduced. Since serum levels of thyroid are the sum of both natural thyroid and thyroid med, as stated in Rec. 10 on p. 13 of the following link, "Only when TSH is no longer stimulating natural thyroid hormone production, or is suppressed, will serum thyroid levels reflect further increases in thyroid medication."
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
I highly recommend reading at least the first two pages of the link and more if you want to get into the discussion and scientific evidence for all that is recommended. You will note that a good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. In fact, as noted in Rec. 10, page 13 of the link, when taking adequate thyroid med, TSH frequently becomes suppressed below range.
As stated by an excellent thyroid doctor, "in tests done about 24 to 28 hrs after their last daily dose, most people on adequate NDT therapy have a suppressed TSH. They usually have FT4 levels that are 1 to 1.3ng/dL, and free T3 levels that are rather high in the range or even slightly above the range. The higher FT3 level compensates for the lower FT4 levels on NDT. These patients have no symptoms or signs of hyperthyroidism--if such occur the
dose is reduced." Note also the deferral of thyroid med until after blood draw to prevent false high results.
Hopefully you can use this info to persuade your doctor to continue to increase your dosage as needed to relieve symptoms, rather than by TSH. Otherwise you will need to find a doctor that will treat clinically, as described. Also hypo patients are frequently deficient in Vitamin D, B12 and ferritin, so you should ask to be tested and then supplement as needed to optimize. D should be at least 50, B12 in the upper end of its range, adn ferritin should be at least 100.