Large language models (LLMs) allow for sophisticated qualitative coding of large datasets, but zero- and few-shot classifiers can produce an intolerable number of errors, even with careful, validated prompting. We present a simple, generalizable two-stage workflow: an LLM applies a human-designed, LLM-adapted codebook; a secondary LLM critic performs self-reflection on each positive label by re-reading the source text alongside the first model's rationale and issuing a final decision. We evaluate this approach on six qualitative codes over 3,000 high-content emails from Apache Software Foundation project evaluation discussions. Our human-derived audit of 360 positive annotations (60 passages by six codes) found that the first-line LLM had a false-positive rate of 8% to 54%, despite F1 scores of 0.74 and 1.00 in testing. Subsequent recoding of all stage-one annotations via a second self-reflection stage improved F1 by 0.04 to 0.25, bringing two especially poor performing codes up to 0.69 and 0.79 from 0.52 and 0.55 respectively. Our manual evaluation identified two recurrent error classes: misinterpretation (violations of code definitions) and meta-discussion (debate about a project evaluation criterion mistaken for its use as a decision justification). Code-specific critic clauses addressing observed failure modes were especially effective with testing and refinement, replicating the codebook-adaption process for LLM interpretation in stage-one. We explain how favoring recall in first-line LLM annotation combined with secondary critique delivers precision-first, compute-light control. With human guidance and validation, self-reflection slots into existing LLM-assisted annotation pipelines to reduce noise and potentially salvage unusable classifiers.