Obviously your Total T3 is terribly low. I expect that the biologically active thyroid hormone Free T3 is also way too low and is the cause for you feeling so bad. Hypothyroidism is correctly defined as "insufficient T3 effect in tissue throughout the body due to inadequate supply of, or response to thyroid hormone". So it is the "TISSUE T3 EFFECT" that determines your thyroid status. Insufficient TISSUE T3 EFFECT results in hypothyroid symptoms.
Most doctors only prescribe T4 meds on the belief that T4 always converts to T3 as needed. Scientific studies have shown this to be incorrect. Studies have also shown that in order to get Free T3 to mid-range or slightly above, as needed by most hypo patients, required T4 doses adequate to drive Free T4 to the top of the range or above. Doctors are reluctant to do this because of their fear that the attendant suppressed TSH means hyperthyroidism so they want to reduce the med dosage. All this is very wrong.
A good thyroid doctor will treat a hypothyroid patient clinically, by testing and adjusting Free T4 and Free T3 (not Total T3) as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results, and especially not TSH. If you want to confirm what I have said, just click on my name and then scroll down to my Journal and read at least the one page Overview of a full paper on Diagnosis and Treatment of Hypothyroidism: A Patient's Perspective.
What dose of T4 were you prescribed? The doctor is correct that it will be weeks before the T4 med has any significant effect. That is because the half-life of T4 is about one week. That means it will be 5 weeks before the T4 reaches over 95% of its final effect on serum T4 levels. Even then it is unlikely that your Free T3 will be adequate to relieve your symptoms. So, if it were me I would ask the doctor to add some T3 med to your dosage. T3 has a half life of less than a day, so you would get some relief much sooner if the doctor gave you an adequate dose of T3 med.
Along with that I would ask the doctor if he is willing to treat clinically, as described, rather than based on TSH levels? For the long term, also ask if the doctor is willing to prescribe T3 meds like NatureThroid and Armour Thyroid and Cytomel. If either answer is not, then you will need to find a good thyroid doctor that will do so.
Also, don't forget the importance of Vitamin D, B12 and ferritin. You should ask for those to be tested also, so you can supplement as needed to optimize.
D should be at least 50 ng/mL, B12 in the upper part of its range, and ferritin should be at least 100.
I said same thing "why" did they delay giving my thyroid meds after my thyroidectomy ....I feel this is to blame for my hideous legs pains....regret not the word for how I feel. I hope you feel better soon