Ferritin is very important for you, so I would try something like VitronC, which has Vitamin C in it to help prevent unwanted effects from the iron. If oral iron supplements don't work, perhaps iron infusion would be an alternative. Whatever it takes, try to get your ferritin optimal.
When you go back for followup tests, I would try hard to get the Reverse T3, and cortisol, along with Free T4 and Free T3. You will need all of those. And please keep in touch and let us know how you are progressing.
Easy to see why you would have hypothyroid symptoms with those levels of Free T4 and Free T3. To relieve symptoms, Free T4 should be at least mid-range, so an increase in T4 med would be advisable, in addition to the T3 already added. In addition, there are other variables that can affect thyroid levels and effects at the cellular level. So I suggest that you should also ask for additional tests for Vitamin D, B12, and ferritin, and then supplement as needed to optimize. D should be at least 50, B12 in the upper end of its range, and ferritin should be at least 70, and some sources say 100. In addition it would be good to test for Reverse T3, and Free T3 from the same blood draw, along with cortisol. You can confirm the need for these tests by reading at least the first two pages of the following link, and more if you want to get into the discussion and scientific evidence supporting all the recommendations.
http://www.thyroiduk.org.uk/tuk/TUK_PDFs/diagnosis_and_treatment_of_hypothyroidism_issue_1.pdf
Regarding your question about the effect of T3, it has a half life of about 1 day. So, it takes about a week to see full effect on serum T3 levels. Symptoms tend to lag somewhat behind changes in thyroid hormone levels. the body needs some time to heal. So I would give it a few more weeks. In the interim you can get the additional testing done and have a clearer picture of all that needs to be done.
Note that members say that their Free T4 needed to be at least mid-range, and their Free T3 in the upper half of the range, and adjusted from there as needed to relieve symptoms. Along with that the other variables I mentioned need to be optimized.