The center of pressure does not move forward when transitioning into supersonic flight, but back. Let me compare the pressure distribution around a sub- and a supersonic airfoil, and the reason should be obvious:
First subsonic flow, showing an Eppler 502 at a moderate angle of attack (3°):
And here is a rhombic airfoil in supersonic flow, copied from this answer:
While the pressure difference between upper and lower side is highest near the leading edge in subsonic flow and tapers off the more you move backward, the pressure difference between both sides is constant over chord in case of the supersonic airfoil. An uncambered rhombic airfoil is the simplest case: The pressure jumps up when the air passes through a shock and jumps down when it passes trough an expansion fan. In between, pressure is constant because the local inclination of the surface does not change.
As a consequence, the center of pressure is near the quarter point in subsonic flow and at mid-chord in supersonic flow (when tip effects are neglected).
Even in transsonic flight, when the local supersonic flow delays the pressure rise, the center of pressure starts to move back. This is the reason for the pitch-down experienced when accelerating above the critical Mach number which is called Mach tuck.
OK, now I forgot to mention the center of drag so far. It is of minor importance because the drag force is so much smaller than the lift force. The concept is the same: Sum up all drag contributions and find the point where a discrete force applied in the direction of drag has the same effect. While the center of pressure just sums up all pressure (and includes the portions which contribute to drag), the center of drag mixes both shear and pressure effects and looks at the fraction which acts in flow direction.