The name means the month of November. Shimotsuki was an Akizuki-class destroyer built by Mitsubishi at Nagasaki. She was laid down on 6 July 1942 and launched on 7 April 1943. In July 1943 the damaged Akizuki reached Nagasaki, and to speed her repairs the forward part of Shimotsuki's hull was cut off at frame 75 and used as Akizuki's new bow. That delayed Shimotsuki; a full new hull was in place by December 1943. She was completed on 31 March 1944 under Lieutenant Commander Kenji Hatano.
She joined Destroyer Squadron 10 on 10 June 1944 and sailed for the Philippine Sea. On 19 June she was the direct escort of Carrier Division 1 and worked on anti-submarine duty while Taiho and Shokaku were torpedoed. On 20 June she claimed two aircraft in the twilight air battle. Near misses gave her steering problems, but she stayed with the main body on the retirement. In July she and Wakatsuki escorted the battleships Kongo and Nagato to Manila.
On 25 November 1944 she was flagship of the 31st Escort Squadron in the South China Sea when an American submarine torpedoed her. The loss of life was very heavy and included Rear Admiral Edo, the squadron commander, his staff and Commander Hatano. Momo picked up survivors; the TROM puts them at 46. Divers found the wreck in 2002 and it was confirmed in May 2003. It lies on its port side, and the damage is consistent with four torpedo hits.
Commanding officers, from the combinedfleet.com record: Lieutenant Commander Kenji Hatano, from 31 March 1944 (lost when the ship was torpedoed and sunk on 25 November 1944, with the squadron staff).
Sources: Allyn Nevitt's combinedfleet.com Tabular Record of Movement for IJN Shimotsuki; Wikipedia, Japanese destroyer Shimotsuki, rewritten here.