I have to disagree with some of what Dallas26 has said. I am not a physician, but I have access through my library to Medline, which is an online database of virtually every article published in the medical journals during the past forty years or so.
I have read many articles on gynecomastia during the past few months, ever since I developed tenderness under my nipples on both sides about six months ago. Such tenderness is very frequently associated with gynecomastia-- especially gynecomastia of recent onset.
The cause is an imbalance of hormones, and one wants to treat the underlying cause, if possible, before proceeding to surgery.
In my case, my doctor examined my breasts and found some "puffiness" under the nipple, especially on the right side. He ordered blood tests to check hormone levels. Those tests showed normal thyroid, but testosterone was in the very low normal range, and LH (Luteinizing hormone) was higher than normal. Although it is rare, high LH with normal testosterone can be a sign of a pituitary tumor. Therefore, my doctor ordered an MRI of the pituitary to rule out that possibility. The MRI showed no abnormalities in the pituitary gland. My doctor also sent me to a general surgeon for his opinion on whether the lumps under my nipples might be malignant. The surgeon said he thought not, but that he could remove the lumps if I wished.
I decided to wait and watch. The lumps are not growing, and the tenderness is less severe than it was a few months ago. I am a lot older than you are (64), and my hormonal changes are probably associated with aging.
I agree with your doctor that it is not normal at your age to develop tenderness and growth of tissue in your breasts. Hyperthyroidism can be a cause. Be patient and let your doctors try to identify the cause. If it is possible to treat the underlying cause, that is a much better option than surgery.
Perhaps the best general article on the subject is "Evaluation and Treatment of Gynecomastia," by Janis F. Newman, in the journal American Family Physician, Volume 55, number 5, April 1997. If you do not have access to Medline, you could go to a university library and probably get access on one of their computers, or you could go the library at a medical school and find a print copy of the journal.
Good luck to you, and keep us posted on what you learn!