Hi and welcome
If you have Hashimoto's you may find that iodine actually does more harm than good unless you know you have a deficiency. Your dietician is also correct, to a degree, about taking iodine and thyroxine. Iodine is used by your body to make thyroxine but, as you are getting some ready-made thyroxine in the form of your tablets, you don't really need to take extra iodine as well.
Thyroxine works well for many people, but for a few it causes nothing but trouble. Those are the people you are likely to hear from on here more often than not. Most people take thyroxine and get well so they don't need sites such as this.
For those of us for whom thyroxine doesn't work well, there are certain things worth looking into before abandoning thyroxine completely. If your adrenals aren't functioning as well as they should be (very common in people with thyroid disorders) your cortisol levels may be too low or too high. This can result in your thyroxine being converted into more of the inactive thyroid and less of the active one that your body uses. It can also result in rather strange over-reactions to very small increases in dosage of thyroxine.
Other things to look at are serum iron, ferritin (stored iron), vitamin B12, folate and vitamin D. Deficiencies, or even just suboptimal levels, of any of these can mean that the thyroid hormones are unable to get into your cells to be used as well as they should be. This can result in them building up in your blood and making you feel quite unwell in various ways, including feeling hyper on a low dose. Deficiencies (or suboptimal levels) also have symptoms very similar to those of hypothyroidism and, while your thyroid treatment may be adequate, leave you with lingering symptoms that you may attribute to your thyroid.
I really would advise getting blood tests for the above mentioned vitamins and minerals. You may be surprised. Don't settle for "normal" as an answer; get the actual numbers. You want them all to be well within the range and most definitely not anywhere near the lower end.
I hope that helps clarify things for you a little bit. You may also find the main Thyroid UK website helpful for information thyroiduk.org.uk/tuk/index....