In a process known as facilitated diffusion, DNA-binding proteins find their target sites by combining three-dimensional diffusion and one-dimensional scanning of the DNA. Following the trade-off between speed and stability, agile exploration of DNA requires loose binding, whereas, at the DNA target site, the searching protein needs to establish tight interactions with the DNA. To enable both efficient search and stable binding, DNA-binding proteins and DNA often switch conformations upon recognition. Here, we study the one-dimensional diffusion and DNA binding of the dimeric lac repressor (LacI), which was reported to adopt two different conformations when binding different conformations of DNA. Using coarse-grained molecular dynamic simulations, we studied the diffusion and the sequence-specific binding of these conformations of LacI, as well as their truncated or monomeric variants, with two DNA conformations: straight and bent. The simulations were compared to experimental observables. This study supports that linear diffusion along DNA combines tight rotation-coupled groove tracking and rotation-decoupled hopping, where the protein briefly dissociates and reassociates just a few base pairs away. Tight groove tracking is crucial for target-site recognition, while hopping speeds up the overall search process. We investigated the diffusion of different LacI conformations on DNA and show how the flexibility of LacI's hinge regions ensures agility on DNA as well as faithful groove tracking. If the hinge regions instead form α-helices at the protein-DNA interface, tight groove tracking is not possible. On the contrary, the helical hinge region is essential for tight binding to bent, specific DNA, for the formation of the specific complex. Based on our study of different encounter complexes, we argue that the conformational change in LacI and DNA bending are somewhat coupled. Our findings underline the importance of two distinct protein conformations for facilitated diffusion and specific binding, respectively.