Although your TSH is high, your Free T4 is just slightly above mid-range, which is about where we'd want it.
The TPOab (Thyroid Peroxidase Antibodies) confirms that you have Hashimoto's.
Have you ever had a Free T3 test? Free T3 is the thyroid hormone that's actually used by individual cells in the body, whereas Free T4 is a storage hormone that must be converted to Free T3 prior to being used. It appears that you might not be converting adequately.
Is the heart palpitations the only symptom you're having? Palpitations can be symptoms of either over or under medication.
If you have a Free T3 test result, please post it, along with the reference range. Also, if you have other symptoms, please let us know what they might be.
Have you been tested for vitamin B-12, D and Ferritin? If you have results for those, also, please post them with reference ranges as many people with Hashimoto's are low in these vitamins/minerals which are necessary for proper thyroid hormone metabolism.
Without a Free T3, we really can't tell what's actually happening since the Free T4 must be converted to Free T3 prior to being used. Can you ask your doctor to test the Free T3?
Are the palpitations the only symptom you're having?
Your doctor is going by the TSH, which is backward from what you'd think - when it's high, thyroid hormone levels are, typically, too low so most people are hypo. When TSH levels are low, thyroid hormone levels are , typically, high and the patient is hyper. TSH is what many doctors go by when determining thyroid status and deciding whether to medicate, increase/decrease, etc. It's important to know, however, that TSH is a pituitary hormone, not a thyroid hormone and there are many circumstances under which it does not correlate well with actual thyroid hormones.
Free T4 and Free T3 should be tested every time TSH is tested. If your doctor refuses to test Free T3, it might be time to look for a new doctor in order to make sure you get the best testing and/or treatment.
Have you been tested for vitamin B-12, D and Ferritin?