The interaction of radio-frequency (RF) waves with edge turbulence modifies the incident wave-spectrum, and can significantly affect RF heating and current drive in tokamaks. Previous LH scattering models have either used the weak-turbulence approximation, or treated more realistic, filamentary turbulence in the ray-tracing limit. In this work, a new model is introduced which retains full-wave effects of RF scattering in filamentary turbulence. First, a Mie-scattering technique models the interaction of an incident wave with a single Gaussian filament. Next, an effective differential scattering-width is derived for a statistical ensemble of filaments. Lastly, a Markov chain solves for the transmitted wave-spectrum in slab geometry. This model is applied to LH launch for current drive. The resulting wave-spectrum is asymmetrically broadened in wave-number angle-space. This asymmetry is not accounted for in previous LH scattering models. The modified wave-spectrum is coupled to a ray-tracing/Fokker-Planck solver (GENRAY/CQL3D) to study its impact on current drive. The resulting current profile is greatly altered, and there is significant increase in on-axis current and decrease in off-axis peaks. This is attributed to a portion of the modified wave-spectrum that strongly damps on-axis during first-pass.