Diffusion language models (Diffusion-LMs) introduce an explicit temporal dimension into text generation, yet how this structure can be leveraged to control generation diversity for exploring multiple valid semantic or reasoning paths remains underexplored. In this paper, we show that Diffusion-LMs, like diffusion models in image generation, exhibit a temporal division of labor: early denoising steps largely determine the global semantic structure, while later steps focus on local lexical refinement. Building on this insight, we propose Time-Annealed Perturbation Sampling (TAPS), a training-free inference strategy that encourages semantic branching early in the diffusion process while progressively reducing perturbations to preserve fluency and instruction adherence. TAPS is compatible with both non-autoregressive and semi-autoregressive Diffusion backbones, demonstrated on LLaDA and TraDo in our paper, and consistently improves output diversity across creative writing and reasoning benchmarks without compromising generation quality.