You need to distinguish between boundary layer flow and outer flow.
Separation means that a parcel of low energy air moves with the wing (here the pressure is about -0.2, expressed as a pressure coefficient). For the outer flow, the contour of the airfoil changes to one of the original airfoil plus the parcel of separated flow. Since this is on the rear upper side of the airfoil, the airfoil "looks" thicker and longer, effectively presenting to the outer flow a body of less camber and angle of attack.
This in turn will reduce the suction in the region of attached flow right past the nose, where normally a high suction peak produces both lift and nose thrust. With a reduced suction peak, the airfoil produces more drag and less lift.
Viscous and inviscid pressure distribution at 12° AoA, calculated with XFOIL 5.4. The dashed lines show the inviscid pressure distribution where flow separation does not happen, while the solid lines show the viscous pressure distribution with separated flow over the last 20% of the upper side. The lines around the airfoil contour show the boundary between the boundary layer and the outer flow. Separation thickens the boundary layer, substantially altering the shape around which the outer flow streams.
The altered airfoil shape causes a reduction in the effective angle of attack, so both the suction on top and the pressure on the bottom are lower. This means that separation causes lift loss; once separation starts, the lift curve slope becomes flatter and can reverse (which indicates a full stall).
In the sample shown above, only on the last 12% of the upper side would separation cause pressure to be lower than in the inviscid case. Note that pressure on the bottom is also lower, so the difference between both (which causes lift) is quite unchanged by separation. Without separation, viscous and inviscid pressure distribution are quite similar, therefore the inviscid distribution is a valid first-order approximation of the viscous pressure distribution without separation.
The reduced suction peak at the nose and the higher suction on those last 12% is what causes more pressure drag. Note that the area around the suction peak points forward and that near the trailing edge points a little backward (especially at high angle of attack). This means that less of the nose thrust pulls the airfoil forward and also there is less pushing from the pressure near the trailing edge.
To estimate the drag increase due to separation of the full airplane configuration, assume a drag increase as if the lift would increase linearly, without separation taking place, including induced drag. Combine this drag with the "real" lift, including losses due to separation, and you will arrive at a surprisingly realistic lift-to-drag ratio of the aircraft between separation onset and stall.