Maintaining engagement in immersive meetings is challenging, particularly when users must catch up on missed content after disruptions. While transcription interfaces can help, table-fixed panels have the potential to distract users from the group, diminishing social presence, while avatar-fixed captions fail to provide past context. We present EngageSync, a context-aware avatar-fixed transcription interface that adapts based on user engagement, offering live transcriptions and LLM-generated summaries to enhance catching up while preserving social presence. We implemented a live VR meeting setup for a 12-participant formative study and elicited design considerations. In two user studies with small (3 avatars) and mid-sized (7 avatars) groups, EngageSync significantly improved social presence (p < .05) and time spent gazing at others in the group instead of the interface over table-fixed panels. Also, it reduced re-engagement time and increased information recall (p < .05) over avatar-fixed interfaces, with stronger effects in mid-sized groups (p < .01).