Multimodal Emotion Recognition in Conversation (ERC) has garnered growing attention from research communities in various fields. In this paper, we propose a cross-modal fusion network with emotion-shift awareness (CFN-ESA) for ERC. Extant approaches employ each modality equally without distinguishing the amount of emotional information, rendering it hard to adequately extract complementary and associative information from multimodal data. To cope with this problem, in CFN-ESA, textual modalities are treated as the primary source of emotional information, while visual and acoustic modalities are taken as the secondary sources. Besides, most multimodal ERC models ignore emotion-shift information and overfocus on contextual information, leading to the failure of emotion recognition under emotion-shift scenario. We elaborate an emotion-shift module to address this challenge. CFN-ESA mainly consists of the unimodal encoder (RUME), cross-modal encoder (ACME), and emotion-shift module (LESM). RUME is applied to extract conversation-level contextual emotional cues while pulling together the data distributions between modalities; ACME is utilized to perform multimodal interaction centered on textual modality; LESM is used to model emotion shift and capture related information, thereby guide the learning of the main task. Experimental results demonstrate that CFN-ESA can effectively promote performance for ERC and remarkably outperform the state-of-the-art models.