Shotgun works great against Phantoms and other strong enemies. You can use it right after a fight has started or after you've slowed down/stopped an opponent with GLOO Cannon.
The shotgun can be found in a security station (screenshot) located on the first floor of Talos I Lobby. You can enter the security room by walking over the yellow tube and then using the small shaft or by using a keycard on the reader by the door. The keycard to the security office can be found in the IT security room on the second level of the lobby (watch out for phantoms!).
Shotgun can also be found in various locations around the station - investigate each area thoroughfully in order to find the weapon.