Generative artificial intelligence (AI) can create sophisticated textual and multimodal content readily available to students. Writing intensive courses and disciplines that use writing as a major form of assessment are significantly impacted by advancements in generative AI, as the technology has the potential to revolutionize how students write and how they perceive writing as a fundamental literacy skill. However, educators are still at the beginning stage of understanding students' integration of generative AI in their actual writing process. This study addresses the urgent need to uncover how students engage with ChatGPT throughout different components of their writing processes and their perceptions of the opportunities and challenges of generative AI. Adopting a phenomenological research design, the study explored the writing practices of six students, including both native and nonnative English speakers, in a first-year writing class at a higher education institution in the US. Thematic analysis of students' written products, self-reflections, and interviews suggests that students utilized ChatGPT for brainstorming and organizing ideas as well as assisting with both global (e.g., argument, structure, coherence) and local issues of writing (e.g., syntax, diction, grammar), while they also had various ethical and practical concerns about the use of ChatGPT. The study brought to front two dilemmas encountered by students in their generative AI-assisted writing: (1) the challenging balance between incorporating AI to enhance writing and maintaining their authentic voice, and (2) the dilemma of weighing the potential loss of learning experiences against the emergence of new learning opportunities accompanying AI integration. These dilemmas highlight the need to rethink learning in an increasingly AI-mediated educational context, emphasizing the importance of fostering students' critical AI literacy to promote their authorial voice and learning in AI-human collaboration.