We present new deterministic algorithms for computing distributed weighted minimum weight cycle (MWC) in undirected and directed graphs and distributed weighted all nodes shortest cycle (ANSC) in directed graphs. Our algorithms for these problems run in $\tilde{O}(n)$ rounds in the CONGEST model on graphs with arbitrary non-negative edge weights, matching the lower bound up to polylogarithmic factors. Before our work, no near linear rounds deterministic algorithms were known for these problems. The previous best bound for solving these problems deterministically requires an initial computation of all pairs shortest paths (APSP) on the given graph, followed by post-processing of $O(n)$ rounds, and in total takes $\tilde{O}(n^{4/3})$ rounds, using deterministic APSP~\cite{AR-SPAA20}. The main component of our new $\tilde{O}(n)$ rounds algorithms is a deterministic technique for constructing a sequence of successive blocker sets. These blocker sets are then treated as source nodes to compute $h$-hop shortest paths, which can then be used to compute candidate shortest cycles whose hop length lies in a particular range. The shortest cycles can then be obtained by selecting the cycle with the minimum weight from all these candidate cycles. Additionally using the above blocker set sequence technique, we also obtain $\tilde{O}(n)$ rounds deterministic algorithm for the multi-source shortest paths problem (MSSP) for both directed and undirected graphs, given that the size of the source set is at most $\sqrt{n}$. This new result for MSSP can be a step towards obtaining a $o(n^{4/3})$ rounds algorithm for deterministic APSP. We also believe that our new blocker set sequence technique may have potential applications for other distributed algorithms.