People starting on thyroid med often find that their biologically active thyroid hormone levels, Free T4 and Free T3, stay the same or even go down slightly. That is because the once a day med dosage causes the TSH to go down and in turn the output of natural thyroid hormone also drops. Since serum FT4 and FT3 levels are the sum of both natural thyroid plus thyroid med, only when TSH is suppressed enough to no longer stimulate natural thyroid hormone production will serum thyroid levels reflect further increases in thyroid medication. So most often we hear that people starting on thyroid med feel no better, or even slightly worse, when first starting on thyroid med.
So it is not really a side effect that your body has to get used to, it is the effect of the med on your TSH and thus the output of natural thyroid hormone. Since we usually only hear of patients starting on a small dose of 25 mcg of T4, or sometimes 50, I wonder also if that dosage causing an immediate drop in TSH would result in an almost immediate drop in thyroid gland output, while it takes 4-5 weeks for the T4 med to reach full effect on serum levels. If so, that might explain the extent of your symptoms. I am going to do some searching on that and will get back to you with any suggestions for short term relief.
In the interim, I think you will benefit from reading at least the first two pages of the following link, and more, if you want to get into the discussion and scientific evidence for all that is recommended. Note especially Rec. 13 on page 13.
I meant to ask what symptoms, if any, did you have prior to starting on the thyroid med?
Not finding anything to confirm my suspicion about the 100 mcg dose causing an immediate suppression of TSH; however, I do believe your 100 mcg starting dose is most likely to have caused a drop in TSH and a drop in your serum Free T4 and Free T3 levels with resulting symptoms you are having. You could confirm this by trying to get the doctor to test for the biologically active thyroid hormones Free T4 and Free T3 (not the same as Total T4 and Total T3). I would also ask to be tested for cortisol, Vitamin D, B12 and ferritin and then supplement as needed to optimize. D should be at least 50 ng/mL, B12 in the upper end of its range, and ferritin should be at least 100. All three are important for a hypothyroid patient. Do you think you can get all 6 of these tests done now?
The other thing to note is that a good thyroid doctor will treat a hypothyroid patient clinically by testing and adjusting Free T4 and Free T3 as needed to relieve symptoms, without being influenced by resultant TSH levels. Symptom relief should be all important, not just test results. You will find all this in the link I gave you above.
You could also consider asking the doctor to reduce your med dosage to maybe 50 mcg and see if it improved your symptoms. Then you could increase your dosage every 4-6 weeks, in maybe 25 mcg increments, as needed to get your FT4 and FT3 levels optimal. Many of us have found that we needed Free T4 at least mid-range and Free T3 in the upper third of the range, and adjusted from there as needed to relieve symptoms.