Yes, usually the "S" is just the wing reference area, and doesn't change.
However, it IS true to say that as AoA varies from the optimum for the body for profile drag, the drag contribution from the body will increase. You can think of it as either a change to profile-Cd or an area change - I prefer the former - but the effect is the same...more drag.
ASSUMING that the lighter aircraft is closer to the optimum AoA, of course ...