Tom,
you are confusing two completely different things :
The yaw damper is not a sideslip angle controler, it compensates movement around the vertical axis, not misalignment with flight direction. The yaw damper is either using the inertial platform or a lateral accelerometer at the nose or the tail of the plane to detect movement around the vertical axis (yaw) and gives adverse rudder to stop this movement. It is designed to avoid dutch rolling of the plane and not to avoid sideslip angles. If the plane is flying stable with 5° sideslip (ie. due to engine thrust asymetry), the yaw damper is not doing anything.
In case of sudden engine failure a yaw will occur and the yaw damper will give adverse rudder, but limited autority will produce a certain sideslip angle. If this angle is stabilized, the yaw damper will not react anymore to the situation.