Online/offline identity-based signature (OO-IBS) is a versatile cryptographic tool to provide the message authentication and integrity in mobile devices, since it lightens the computational burden after the signer receiving the message and eliminates the overhead of certificate management. It has several valuable applications, for instance, wireless sensor networks. Identity-based chameleon hash (IB-CH), as an alternative building block to construct OO-IBS, has been explored in numerous literatures. Nevertheless, there still exist two major issues. 1) Nearly all of the previous IB-CH schemes with weak collision-resistance (W-CollRes) are with random oracles, which may lead to security risks in practicality. The only IB-CH scheme in the standard model suffers from the large size of public parameters and inefficient setup process. 2) The only IB-CH scheme without key exposure also relies on random oracles. In this paper, we propose two novel IB-CH schemes in the standard model. The first scheme is adaptive identity, W-CollRes secure and efficient, significantly reducing the computation costs of all algorithms and the size of public parameters compared with the existing scheme in the standard model. The second scheme is the first IB-CH achieving key exposure freeness without random oracles. Both theoretical and experimental analyses demonstrate the good performance of our proposed schemes. Furthermore, we apply our schemes to optimizing the existing generic OO-IBS construction. The optimized generic constructions reduce computational overhead by 50.0% in the online phase and enable the hash value/signature tuple generated in the offline phase to be reusable, respectively.