My dentist said that thyroid problems could be the cause of my dental problems and that I should have it corrected soon. My physician says she wants to want till my dental problems are corrected before possible starting me on medication. There are very few reasons to put off starting thyroid replacement. About the only one with any validity is a patient with angina who will have more angina with thyroid replacement. I can really think of no reason why your physician would put off replacing your thyroid if it is indeed low.
Hypothyroidism can be due to glandular failure or to failure of the pituitary gland.
With your menopausal symptoms, I would be absolutely sure that your pituitary is functioning. In general, a good family practice physician, internist or endocrinologist could do these tests. If it is only your thyroid any of the three could replace the hormone. If the pituitary is not functioning correctly, I'd see an endocrinologist. My mother was an extremely healthy, self-sufficient 85 year old.