Those side effects you mention are not from the Eltroxin, but typically from an inadequate dosage, which leaves the patient hypothyroid due to low levels of Free T4 and Free T3. If her TSH remains high while on the Eltroxin, that is another sign of being under medicated.
You cannot know for sure about the need for the med without adequate testing. Since you mention a high TSH, I'd start with the assumption that she is hypothyroid because of Hashimoto's Thyroiditis and request that she be tested for Free T4, Free T3, TSH, Thyroid Peroxidase and Thyroglobulin antibodies (TPO ab and TG ab), Reverse T3, morning serum cortisol, Vitamin D, B12 and ferritin. If there is resistance to doing all those tests, you should mention all the symptoms that she has and insist the tests are needed to diagnose her status.
Once you have those test results we can better assess her status and advise further.
She did mention her doctor wants to higher the dose. She is taking 100mcg and once a week adds 50mcg with the 100.
Her T4 is around the 13 range (10-20) and TSH is 6/7 (0.40-4.20), but these were done last year.
She will be doing her blood work soon anyways so she will request to add the ones you metioned, I don't think they will refuse.
Will update when all is done, thank you!