During quadrotor load transport, the cable’s elasticity exacerbates load fluctuations, which may result in platform instability or a potential crash. This paper introduced a model of the connecting cable as a spring-damper system and established the dynamic model of the suspension system based on Newton’s law. Nonsingular fast terminal sliding mode control (NFTSMC) was employed for attitude, position, and anti-swing controller design. Adaptive controllers were integrated into altitude control to address uncertainties related to load mass and cable length. The inclusion of an anti-swing controller into the position control loop effectively dampens load oscillations while ensuring accurate position tracking. Numerical simulations demonstrated that the proposed controller outperforms both the energy-based controller and the conventional linear sliding mode controller.