Nodules do not cause hypothyroidism -- unless the nodule(s) replace the entire normal thyroid structure -- which is unlikely in the case of a single nodule.
US guided FNA is easy to do - the thyroid itself is next to the trachea and esophagus, so the nodules are often wedged in this 'tracheo-esophogeal groove' -- but still safe to FNA with a small needle using ultrasound.
Synthroid is no longer typically used to treat nodules unless there is concomitant hypothyroidism for other reasons (most common would be hashi, but your AB's are negative).
Sounds like the 2nd op is a good move. The other option is to surgically remove it as you seem to have some compressive symptoms. BUT, a good surgeon will want an FNA first in order to determine the extent of surgery.