Masked diffusion language models (DLMs) generate text through iterative parallel decoding, but at low step budgets, committing many tokens independently can produce premature termination or repetitive outputs. Continuous embedding-space denoising can mitigate this failure mode by maintaining soft states for all positions, yet training such models from scratch at scale remains challenging. We show that a pretrained masked DLM can instead be efficiently adapted to support continuous denoising. Starting from LLaDA-8B-Instruct, we perform only 1,000 steps of continued pretraining with Discrete Stochastic Localization (DSL), replacing binary masking with continuous per-token Gaussian noise mapped to soft embedding mixtures. The adapted model supports SDE-based inference that evolves all positions jointly and defers hard token commitment to the final step. At ≤16 denoising evaluations, DSL-LLaDA substantially mitigates the premature-termination and repetition tradeoff in open-ended generation and achieves the best ROUGE-1 among the evaluated diffusion baselines on all four zero-shot summarization benchmarks. The same adaptation also enables selective correction of corrupted tokens while preserving more than 98% of clean tokens. Same-compute controls indicate that these behaviors arise from continuous-noise exposure rather than additional training alone.