We present an algorithm for finding a perfect matching in a $3$-edge-connected cubic graph that intersects every $3$-edge cut in exactly one edge. Specifically, we propose an algorithm with a time complexity of $O(n \log^4 n)$, which significantly improves upon the previously known $O(n^3)$-time algorithms for the same problem. The technique we use for the improvement is efficient use of cactus model of 3-edge cuts. As an application, we use our algorithm to compute embeddings of $3$-edge-connected cubic graphs with limited number of singular edges (i.e., edges that are twice in the boundary of one face) in $O(n \log^4 n)$ time; this application contributes to the study of the well-known Cycle Double Cover conjecture.