Social media fact-checking is increasingly challenged by evidence-level and aggregation-level conflicts, where erroneous evidence often mimics authoritative news sources. To support conflict verification task, we construct ContraNote, a real-world dataset derived from X Community Notes. ContraNote is curated to mirror the two aforementioned challenges, providing 33,686 posts for evaluating evidence-level conflict resolution, and 54,474 instances for evaluating aggregation-level evidence prioritization. This paper further proposes CoVer, a framework designed to resolve these contradictions through factual adjudication. Cover uses a three-stage pipeline consisting of evidence schema normalization, factual consensus and support verification, prioritizing definitive evidence over noise to prevent it from compromising the final verdict. Technical evaluations across multiple benchmarks show that CoVer outperforms state-of-the-art baselines, achieving an accuracy of 88.4% on CONFACT-HumC and 89.4% on CONFACT-ModC.