Sampling a random permutation with restricted positions, or equivalently approximating the permanent of a 0-1 matrix, is a fundamental problem in computer science, with several notable results achieved over the years. However, existing algorithms typically exhibit high computational complexity. Achieving the optimal running time remains elusive, even for nontrivial subsets of the problem. Furthermore, existing algorithms primarily focus on a single permutation, leaving many combinatorial problems involving multiple constrained permutations unaddressed. For a single permutation, we achieve the optimal running time $O(n^2)$ for approximating the permanent of a very dense $n \times n$ 0-1 matrix, where each row and column contains at most $\sqrt{(n-2)/20}$ zeros. This result serves as a fundamental building block in our sampling algorithm for multiple permutations. We further introduce a general model called permutations with disjunctive constraints (PDC) for handling multiple constrained permutations. We propose a novel Markov chain-based algorithm for sampling nearly uniform solutions of PDC within a lopsided Lovász Local Lemma (LLL) regime. For uniform PDC formulas, where all constraints are of the same width and all permutations are of the same size, our algorithm runs in nearly linear time with respect to the number of variables. Previous approaches for sampling LLL relied on the variable model. In contrast, the sampling problem of PDC encounters a fundamental challenge: the random variables within each permutation in the joint probability space are not mutually independent, leading to long-range correlations. To tackle this challenge, we introduce a novel sampling framework called correlated factorization and a new concept in the path coupling analysis, termed the inactive vertex.