In this paper, we study the numerical discretization of stochastic differential equations with locally Lipschitz, super-linearly growing drift, and the resulting implications for sampling from non-log-concave distributions satisfying a logarithmic Sobolev inequality. In this regime, the classical Euler--Maruyama scheme underlying the unadjusted Langevin algorithm (ULA) is known to be unstable. We analyze the KL-accelerated tamed unadjusted Langevin algorithm (kTULA) and introduce a new tamed randomized midpoint scheme, termed tRLMC. Building on the shifted-composition approach of \cite{chewi2024local}, we develop two new local-error frameworks that yield finite-time, non-asymptotic error estimates against the underlying SDE -- in KL divergence for kTULA, and in total variation for tRLMC -- valid for general locally Lipschitz drift. Specializing these frameworks to the sampling problem under a logarithmic Sobolev inequality, we obtain a near-optimal $\widetilde{O}(\varepsilon^{-1/2})$ iteration complexity for kTULA in KL divergence, with corresponding guarantees in total variation and Wasserstein distance. We further establish, for the first time, a non-asymptotic guarantee in total variation for a tamed randomized Langevin scheme under super-linear drift growth, together with the corresponding Wasserstein-distance bound, both with $\widetilde{O}(\varepsilon^{-1})$ complexity for tRLMC. As a consequence, both schemes yield non-asymptotic bounds for a non-convex excess-risk optimization problem.