As you mentioned, the main rotor thrust must support the weight of the helicopter in hover, and must also provide a propulsive force to overcome drag in forward flight. However, the induced drag is also a function of how the air is deflected downward by the rotor. The velocity of this downward deflection of air at the rotor is often called induced velocity. This induced velocity is a function of not just thrust, but also the helicopter forward speed, rate of climb, air density, height above ground (when near the ground it decreases, a phenomena known as ground effect), etc.
Relevant to your question is how the induced velocity decreases as the helicopter goes from hover to moderate forward speed (even though thrust increases). At hover, the rotor is sitting in its wake with a large induced velocity. With some speed, the helicopter is moving away from it's wake, resulting in a smaller induced velocity. This smaller induced velocity increases the angle of the incoming air to the blade, and therefore decreases induced drag (and even overall power required).