C-type shocks are believed to be ubiquitous in turbulent molecular clouds thanks to ambipolar diffusion. We perform a linear analysis and reveal a fluid instability in both perpendicular and oblique C-shocks (Gu & Chen 2020; Gu 2021). This new instability in C-shocks, speculated and named the drag instability by Gu et al. (2004), is a local linear overstability phenomenon due to the ion-neutral drag and cosmic-ray ionization. The approximate dispersion relations of the instability are derived, and the most unstable modes are identified. We shall also present the numerical results of the drag instability in 1D C-shocks from non-ideal magnetohydrodynamic simulations in comparison with the local linear theory (Gu et al. 2022). We confirm the presence of the drag instability in simulated 1D isothermal C-shocks. Furthermore, we find that the nonlinear phase of the instability is subject to wave-steepening, leading to saturated perturbation growth. Our study suggests that in the absence of any drivers, the turbulence in star-forming clouds may decay faster than it was previously thought.