Another, potentially faster, way of doing this, without any splits or merges, would be based on pairing polygons.
Say you want to align polygons in the 'red' layer to the 'green' layer (for lack of better terminology) one way is to find a label point for each green and red polygon (Feature to Point, inside), for every red point (corresponding to a red polygon) find the nearest green point and match the corresponding polygons (red polygon to the green polygon which has a label point closest to the red polygon's label point) and then transfer the geometry of the green polygon to the red polygon.
For safety you can verify that the mapping is one to one (no to red polygons choose the same green one) and flag any red polygons that match to a conflicting green polygon (you could merge them if desired). If the shifts are large, the label points may shift too much and this won't work well, but if it works it is considerably less work than a bunch of splits & merges.
