With the low FT4, medication was certainly in order, but I can't imagine why you'd have been put all the way up to 200 mcg. That's a right, hefty dose.
Get a different doctor....
Where you self medicating....???
No you should have not been medicated.....
I would be very doubtful that your current symptoms are due to the lingering effects of the Synthroid. Based on the low Free T4 test result, I would be much more suspicious that the your original diagnosis of hypothyroidism was correct and that you now need to be back on the right dosage of thyroid meds. The best way to find out is to test for the biologically active thyroid hormones, Free T3 and Free T4. While you are at it, I would also suggest testing for Vitamin A, D, B12 and ferritin. Also, since Hashimoto's Thyroiditis is the most common cause of diagnosed hypothyroidism, I would also suggest testing for the thyroid antibodies. Those tests are TPO ab and TG ab.
If you will get those tests done and post results and their reference ranges shown on the lab report, members will be glad to help interpret and advise further.