Some general comments, not to be construed as personal advice, offered here to dispel errors.
Both Hashi's and Graves' are autoimmune disorders. Each has their characteristic lab profile. A solitary TSH, Thyroid Stimulating Hormone level, is actually a hypothalamic hormone, NOT produced by the thyroid. TSH is used alone as a screening test, but in no way gives a complete picture. Clinically there are many similarities between Hashi's and Graves'. Either can exhibit an under-active or over-active thyroid picture. Typically both have a spectrum of presentation and wax and wane over the years. The individual's presentation usually depends upon which phase the patient is in, hypo- or hyper-thyroid. The initial period of thyroid stimulation may be a very mild hyperthyroidism and never detected (unless lab tests are done OR until the thyroid becomes burned out and a hypothyroid symptoms predominate) or dramatic and potentially fatal ("thyroid storm").
Severely hypothyroid patients can have heart failure of the low-output type ( a flabby heart, if you will). Severely hyperthyroid patients can have heart failure of the high-output type (rapid heart, high volume output, if you will). Neither of these situations is good for the patient. It used to be the case that doctors would do frequent testing and adjustment of medications (whether anti-thyroid drugs like methimizole or thyroid replacement, the actual thyroid hormones). Sometimes a partial thyroidectomy or low-dose Iodine-131 partial ablation would be done in hopes of returning hyperthyroid function to normal, but that was unpredictable and rarely successful in the long run because of the waxing and waning of the autoimmune phenomena.
Over the last few decades doctors have come to realize that a favorable outcome for the patient is best accomplished by definitive treatment. This may be why your doctor may recommend surgical or radio-ablation (Iodine-131) of your thyroid followed by thyroid replacement when thyroid hormone levels (T4 and T3) fall. Again, overall thyroid complications are fewer with definitive treatment rather than fiddling around chasing ever-changing doses over the years.
The actual thyroid hormones "T4" (has 4 iodine atoms per molecule, less potent) and "T3" (has 3 iodine atoms per molecule, more potent) are specific molecules that can be obtained synthetically or "naturally" in dried-out pig thyroid glands. While I see significant benefits to certain "organic" and "natural" approaches, when it comes to thyroid hormones, I think that the insistence on "natural" thyroid is a fetish, nothing more.
Lastly, fine tremor is quite common in hyperthyroidism whether from Hashi's or Graves'. The fine tremor of hyperthyroidism is quite distinct in appearance from the rest tremor of Parkinsonism ("pill rolling") or intention tremor of cerebellar disease.