Satisfiability (SAT) with cardinality constraints arises naturally in many practical applications, where high-level counting requirements coexist with standard Conjunctive Normal Form (CNF) clauses. Translating these constraints into CNF can destroy structural information, limiting the effectiveness of search-based heuristics. In this paper, we propose a cardinality-aware local search framework to solve this problem, denoted as CardSAT-LS. CardSAT-LS integrates a preprocessing phase based on the generalized unit propagation and resolution, a cardinality-sensitive scoring function combining the make-break mechanism and cardinality violation, and an initialization based on fake-backbone variables. Furthermore, CardSAT-LS employs a unified framework that adaptively alternates between flip and swap operators when the search gets trapped in local optima. Finally, we conduct experiments on five public benchmarks from real-world applications as well as the MaxSAT and SAT competitions. Compared with ten state-of-the-art competitors, including SAT, MaxSAT, and PB solvers, CardSAT-LS solves the most instances with the lowest PAR-2 score. Additionally, we integrate CardSAT-LS into exact solvers for phase selection, which leads to significant speedups.