Deep Learning (DL)-based channel state information (CSI) feedback is a promising technique for the transmitter to accurately acquire the CSI of massive multiple-input multiple-output (MIMO) systems. As a critical concern about DL-based physical layer application, although most existing CSI feedback methods have shown advantages in dedicated channel environments and scenarios, the generalization and adaptivity of these methods remain challenging. Therefore, we propose a Hybrid Complex-Valued Lightweight framework, namely HybridCVLNet, with a hybrid structure, task, and codeword and correspondent domain adaptation scheme to overcome the data drift and dataset bias for CSI feedback. The experiment verifies the validity of the proposed lightweight HybridCVLNet regularization to the regression process. It achieves stable generalizability and performance gain over the SOTA feedback schemes in an intra-domain multi-category setting. In addition, its hybrid domain adaptation scheme is more efficient and superior to the online direct-finetune method under the unseen cross-domain dataset.