Sampling from heavy-tailed and multimodal distributions is challenging when neither the target density nor the proposal density can be evaluated, as encountered in α-stable Lévy-driven fractional Langevin algorithms. While the target distribution can be estimated from data using score-based or energy-based models, the α-stable proposal density and its score are generally unavailable, rendering classical density-based Metropolis-Hastings (MH) corrections impractical. Consequently, existing fractional Langevin methods operate in an unadjusted regime, often exhibiting substantial finite-time errors and poor empirical control over tail behavior. To address these limitations, we first introduce a general, fully score-based MH adjustment mechanism. Building upon this framework, we develop the Metropolis-Adjusted Fractional Langevin Algorithm (MAFLA). MAFLA employs designed proxies for fractional proposal score gradients under isotropic symmetric \alpha-stable noise and learns the acceptance function via Score Balance Matching. We empirically demonstrate the strong performance of MAFLA across a series of tasks, including combinatorial optimization problems, where the method significantly improves finite-time sampling accuracy over unadjusted fractional Langevin dynamics.