When you move the cyclic forward the pitch of the retreating blade is increased, increasing lift and drag, so that the tip plane is higher for that blade when it reaches the aft position. The advancing blade pitch is reduced to lower the tip plane path, decreasing the lift and drag, so that the tip plane path is lower at the front. The total lift vector is displaced forward but doesn’t change so the rotor drag also doesn’t change. The horizontal component of the thrust vector is increased, causing the acceleration, but the vertical component is reduced. This is where a collective increase is required to maintain level flight and this results in the increased drag/torque. With no collective input the aircraft would accelerate but descend. Due to this acceleration the total aircraft drag is increased but that’s not what’s being discussed here.
As I mentioned above, flight control systems usually have some form of control mixing to accomplish this adjustment with minimal pilot input. This can be mechanical linkages, fly by wire programming or hydro-mechanical, referred to as a mixing unit.