The Sawed-Off Shotgun does not feature blue lights like other COG and Locust weapons, and so could be a Stranded gun used by Gears. It also has more of a kick than the Gnasher, and more power as is seen in the Ashes to Ashes trailer and the Campaign Demo, where Marcus Fenix used it to kill a Drone with one shot and fire at Lambent tentacles respectively.
In the Ashes to Ashes trailer, the weapon fires multiple shots without reload, despite having a one-shot reload in-game. This was later confirmed as artistic licence.
This gun was introduced to the universe because Cliff Bleszinski wanted to embrace the shotgun and give more of a possible selection to selecting a standard load-out for multiplayer in Gears 3.
Lee Perry explained the Sawed-Off’s range capabilities like this: Sawed-Off –> Gnasher –> Retro Lancer –> Lancer –> Hammerburst, showing that the Sawed-Off will be a short range power packed version of the Gnasher. (-Gears Of War Wikia)