First thing to note when taking thyroid medication, TSH should not be used to determine dosage. That is because our bodies evolved with a continuous low flow of thyroid from the gland resulting in an equilibrium among FT4, FT3, and TSH. . So when taking adequate replacement thyroid med in only one or two doses, the equilibrium is totally changed and most patients find that their TSH becomes suppressed. Many doctors don't understand this and erroneously interpret a suppressed TSH as meaning hyperthyroidism for both the untreated and treated states. In reality a suppressed TSH does not mean hyperthyroidism, unless there are accompanying hyper symptoms due to excessive levels of Free T4 and Free T3.
So any effort to target a level of TSH within range when taking thyroid med will usually result in inadequate treatment, like yours.. Although you did not provide reference ranges it sounds like your FT4 is around mid-range, which is adequate, but your FT3 at the low end of the range is inadequate. Note the following conclusion from a recent, excellent scientific study: "Hypothyroid symptom relief was associated with both a T4 dose giving TSH-suppression below the lower reference limit and FT3 elevated further into the upper half of its reference range."
The reason your FT3 is affected by your TSH is that TSH is one of the variables that affect conversion of T4 to T3. People taking thyroid med frequently find that their FT3 level lags their FT4 level, and they have to add a source of T3 to get their FT3 level adequate.
So you need a good thyroid doctor that will treat clinically by testing and adjusting FT4 and FT3 levels as needed to relieve symptoms, rather than based on biochemical tests, and especially not TSH. In addition hypo patients are frequently deficient in Vitamin D, B12 and ferritin. All three are important and should be supplemented as needed to optimize. D should be at least 50 ng/mL, B12 in the upper part of tis range, and ferritin should be at least 100. If you want to confirm what I have said, click on my name and then scroll down to my Journal and read at least the overview of a paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.
"T3 levels are not appropriate for diagnosing hypothyroidism because increased conversion of T4 to T3 maintains T3 serum levels within the normal range until hypothyroidism becomes severe (52) (D)." - The clinical use of thyroid function tests -Arq Bras Endocrinol Metab vol.57 no.3 São Paulo Apr. 2013.
Without going into why this may not be the case for some people, increased conversion might explain why T3 increases when your TSH is rising.