Is it true that In a binary system, the more massive stars always stay closer to the center of mass and move slower?
Yes, the more massive star of a binary system is closer to the centre of mass.
In a binary system, the two bodies always orbit around the centre of mass of the system which is called the barycentre.
At any point in time if the distance between the two bodies is
If the larger object is significantly larger than the smaller object the barycentre is inside its body and the object appears to wobble rather than orbit.