I don't really understand your comment about formulas for drag "without thrust" (and presumably, other formulas for drag "with thrust"). If you can give a link or a picture that might help.
The thrust and drag forces are independent of each other. Thrust comes from the engines. Drag is caused by the air flow over the plane and depends on the air speed and the angle of attack of the wings.
In the question you attached, the plane is flying straight and level so thrust = drag. You can work out the thrust from the change in momentum of the air by the propeller. Them you can find the drag coefficient from the drag force, wing area, air speed, and air density.
In real life, the propeller wash might have some effect on the drag coefficient. For a multi-engine propeller driven plane, the air flow over the wings would be slightly faster than the airspeed of the plane, especially at low airspeed and high engine power. (For a single engine plane the effect would probably be less). But I think that sort of effect would be too complicated to include in a "formula".