Do check these, Vitamin D3 level and Free T3.
Also make sure you get adequate night sleep. Never cut on sleep and also make sure its quality is good, many of the symptoms you are experiencing especially lightheadness can result from poor sleep patterns.
From your symptoms and being diagnosed as having Hashi's, I'd say that you are still very hypothyroid. Even though your Free T4 was near the middle of the range, I expect that your Free T3 level, which was not tested, is inadequate. T4 has to be converted by your body to T3 in order to be metabolized in all the cells of the body. Studies have shown Free T3 to correlate best with hypo symptoms. You need to make sure they always test for both Free T4 and Free T3 every time you go in for tests. Many doctors will tell you that you don't need to test for Free T3, since the body will convert T4 to T3 as needed. That is not true. There are many factors that affect conversion, including ferritin, and your ferritin level is way too low.
A good thyroid doctor will treat a hypo patient clinically by testing adjusting Free T4 and Free T3 levels as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. I say that about TSH levels because other studies have shown that hypo patients frequently find that their TSH level becomes suppressed when taking adequate thyroid med. That doesn't mean hyperthyroidism unless there are accompanying hyper symptoms due to excessive levels of Free T4 and Free T3.
You can read about all this in the following link. I highly recommend reading at least the first two pages and even further if you want to get into the discussion and scientific evidence supporting all that is suggested on page 2.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
So, you first need to do some additional testing. Specifically I recommend three additional tests suggested on page 2 of the link: Free T3, Reverse T3, and Vitamin D, I also suggest that you supplement as needed to optimize Vitamin D, B12 and ferritin. D should be at least 50, B12 in the upper end of its range, and ferritin at least 70, and some say 100. D3, B12 and ferritin supplements are readily available over the counter. For ferritin I recommend Vitron C, to avoid stomach upset from the iron.
With Hashi's the destruction of your thyroid gland will continue, and will require increasing doses of thyroid med to off set the loss of natural thyroid hormone. If tests show your current Free T3 level is in the low end of its range, you should ask your doctor for some T3 med. Just increasing your T4 med is unlikely to provide enough T3 through conversion. If your doctor has a problem with the additional testing or with the T3 med, I suggest you ask him to read a copy of the link and then ask him if he is willing to treat clinically, as described. If not, then you will need to find a good thyroid doctor that will do so. For reference, most of us needed for Free T4 to be at least mid-range, and Free T3 in the upper half of the range, adjusted as needed to relieve symptoms.
You definitely have symptoms that indicate you are still hypothyroid. Before discussing it further please clarify if your blood test results were from before or after starting on the Levo>