This paper focuses on the distributed state estimation (DSE) problem in sensor networks through the application of a consensus-based methodology. In practical scenarios, the estimation errors within the sensor network exhibit inherent correlation, primarily arising from the consideration of same prior estimate and process noise in a distributed estimation system. However, prevailing distributed approaches, particularly those embedded within consensus-based DSE frameworks, simply assume independence among estimation errors or utilize a suboptimal strategy that neglect cross-covariance matrix, which may lead to a degraded estimation results. To address this limitation, we propose the fusion algorithm Weighted Average Consensus considering Correlated Estimates Errors, abbreviated as WACCEE. The algorithm improves DSE performance by integrating the correlation among local estimation errors as indicated by the cross-covariance matrix into the fusion process. Furthermore, we prove that the proposed WACCEE fusion algorithm exhibits consensus. Additionally, the stability of the WACCEE filters is guaranteed through an analysis of mean-square exponential boundedness of the estimation error using stochastic stability theory. Finally, the effectiveness of the proposed algorithm substantiated by its application to a target tracking case study and a connected undirected sensor network.