Translating software written in C to Rust has significant benefits in improving memory safety. However, manual translation is cumbersome, error-prone, and often produces unidiomatic code. Large language models (LLMs) have demonstrated promise in producing idiomatic translations, but offer no correctness guarantees. We propose SACTOR, an LLM-driven C-to-Rust translation tool that employs a two-step process: an initial "unidiomatic" translation to preserve semantics, followed by an "idiomatic" refinement to align with Rust standards. To validate correctness of our function-wise incremental translation that mixes C and Rust, we use end-to-end testing via the foreign function interface. We evaluate SACTOR on 200 programs from two public datasets and on two more real-world scenarios (a 50-sample subset of CRust-Bench and the libogg library), comparing multiple LLMs. Across datasets, SACTOR delivers high end-to-end correctness and produces safe, idiomatic Rust with up to 7 times fewer Clippy warnings; On CRust-Bench, SACTOR achieves an average (across samples) of 85% unidiomatic and 52% idiomatic success, and on libogg it attains full unidiomatic and up to 78% idiomatic coverage on GPT-5.