Use “Split surface by isocurve” (right click on the icon) on the surface you need.
On a polysurface, explode before and join everything after you splitted it.
This is possible, but it’s not really the way rhino was meant to be used.
Dragging edges or control points of solids will likely end up making errors on complex geometries.
Anyway, usually i explode a polysurface and group it… then using solid control points is much safer, if a surface will go crazy by moving points it will not affect other surfaces.