Channel reconstruction and generalization capability are of equal importance for developing channel estimation schemes within deep learning (DL) framework. In this paper, we exploit a novel DL-based scheme for efficient OFDM channel estimation where the neural networks for channel reconstruction and generalization are respectively designed. For the former, we propose a dual-attention-aided super-resolution neural network (DA-SRNN) to map the channels at pilot positions to the whole time-frequency channels. Specifically, the channel-spatial attention mechanism is first introduced to sequentially infer attention maps along two separate dimensions corresponding to two types of underlying channel correlations, and then the lightweight SR module is developed for efficient channel reconstruction. For the latter, we introduce continual learning (CL)-aided training strategies to make the neural network adapt to different channel distributions. Specifically, the elastic weight consolidation (EWC) is introduced as the regularization term in regard to loss function of channel reconstruction, which can constrain the direction and space of updating the important weights of neural networks among different channel distributions. Meanwhile, the corresponding training process is provided in detail. By evaluating under 3rd Generation Partnership Project (3GPP) channel models, numerical results verify the superiority of the proposed channel estimation scheme with significantly improved channel reconstruction and generalization performance over counterparts.