A longstanding challenge in mental well-being support is the reluctance of people to adopt psychologically beneficial activities, often due to a lack of motivation, low perceived trustworthiness, and limited personalization of recommendations. Chatbots have shown promise in promoting positive mental health practices, yet their rigid interaction flows and less human-like conversational experiences present significant limitations. In this work, we explore whether the anthropomorphic design (both LLM's persona design and conversational experience design) can enhance users' perception of the system and their willingness to adopt mental well-being activity recommendations. To this end, we introduce Sunnie, an anthropomorphic LLM-based conversational agent designed to offer personalized guidance for mental well-being support through multi-turn conversation and activity recommendations based on positive psychological theory. An empirical user study comparing the user experience with Sunnie and with a traditional survey-based activity recommendation system suggests that the anthropomorphic characteristics of Sunnie significantly enhance users' perception of the system and the overall usability; nevertheless, users' willingness to adopt activity recommendations did not change significantly.