Accent conversion (AC) in this work refers to converting speech from unseen source speakers to a fixed target accent while preserving linguistic content and speaker identity, without test-time target reference speech, matching-pool reconstruction, gradient updates, or speaker-specific fine-tuning. Compared with fully parametric generation-based AC methods, retrieval-based methods can reuse acoustic representations from real target-accent speech and are therefore promising for preserving natural spectral details. However, existingretrieval-based AC methods are affected by incomplete phoneme coverage in the matching pool, fixed-neighborhood retrieval, and coarse-resolution style modulation. To address these issues, we propose AA-kNN-AC, a retrieval-based framework that improves kNN-based accent conversion through three components: phoneme-coverage-aware matching-pool construction, anisotropy-aware adaptive kNN retrieval, and a progressive AdaIN-based decoder. The matching-pool strategy selects target-accent utterances according to phoneme diversity and rare-phoneme coverage, providing a more balanced target-accent representation for retrieval. The retrieval mechanism adjusts the neighborhood size according to local feature-space anisotropy estimated with singular value decomposition, improving the adaptability of feature matching across different speech regions. The decoder couples temporal up-sampling with AdaIN-based speaker-content fusion to improve spectral continuity and fine-grained acoustic reconstruction. Experimental results on L2-ARCTIC and VCTK show that AA-kNN-AC improves intelligibility, speaker similarity, and naturalness over retrieval-based baselines and remains competitive with representative generative systems under the fixed target-accent matching-pool setting.