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Modern autoregressive ASR systems can emit timestamps as decoded tokens, enabling timestamped transcription without frame-level aligners or inference-time post-processing. We show that these generated timestamps can drift across long non-speech spans: the transcript may remain plausible, but the decoded time axis drifts away from the audio. We study this non-speech-induced timestamp drift with self-built gap and long-gap benchmarks across 15 evaluated timestamp-producing ASR and audio-language systems. Naive timestamp-corrected fine-tuning improves alignment but can severely degrade non-target ASR behavior, exposing a forgetting problem. We propose REDDIT(REplay-based Distribution eDITing), a lightweight two-stage post-training framework that corrects timestamps while avoiding this catastrophic forgetting: it first edits timestamp targets under the model's own replayed decoder context while matching the frozen base distribution on non-timestamp tokens, then applies a short edited-prefix refinement stage. In this framework, we construct correction supervision without human transcripts or human timestamp annotations by combining VAD-trimmed speech spans with inserted non-speech gaps and known concatenation offsets. On Whisper-tiny, 34.9 hours of targeted correction audio used and only 1.6% of model parameters updated, raising long-gap mIoU from 38.7% to 95.0% and reducing mixed-gap out-of-domain AAS from 2752 ms to 223 ms while preserving CV-en MER at 41.3% (versus 524.2% for ordinary SFT decoder tuning).