This randomized phase II trial studies how well thyroid gland removal with or without central lymph node dissection works in treating patients with thyroid cancer or suspected thyroid cancer that has not spread to the lymph nodes. Currently, the standard treatment for thyroid cancer is total thyroidectomy, or complete removal of the thyroid. Sometimes, the lymph nodes in the central part of the neck may also be surgically removed, called central lymph node dissection. Prophylactic removal of the lymph nodes may increase the risk of life-threatening complications, and may reduce post-surgery quality of life. It may also prevent the cancer from returning and reduce the need for additional surgery. It is not yet known whether recurrence rates and complication levels are lower after thyroid gland removal alone or with central lymph node dissection.