No, because drawing your whip again at the end of the turn is not part of your move or your action. For example, you could open a door during your move as you stride toward a foe, or you could draw your weapon as part of the same action you use to attack.

Is stowing a weapon an action?

You can draw/stow one weapon as a free action normally. If you were to draw/stow a second weapon it would fall under the Use Object Action. With the Dual Wielding feat it allows you to draw/stow 2 weapons, making a third weapon fall under the Use Object Action.

Does drawing a weapon count as an action?

No, drawing a weapon does not take an action. Nor does picking up a dropped weapon.