Incomplete transfer of clinical information during nursing handoffs can allow unresolved patient-care concerns to persist across shifts, delay escalation of deteriorating conditions, and create uncertainty about whether identified risks have been appropriately addressed. This study proposes HandoffCARE-Net, a Handoff Continuity, Alerting, Resolution, and Escalation Network, as an artificial intelligence framework for automatically analysing sequential nursing handoff records and identifying clinically important information that remains unresolved. HandoffCARE-Net combines a clinical transformer-based text encoder, cross-handoff temporal attention, patient-concern state memory, risk-sensitive escalation prioritisation, and a closure-verification layer within a unified multi-task learning architecture. The model simultaneously performs unresolved-concern detection, information-continuity classification, escalation-priority prediction, concern-state tracking, and closure-status verification across consecutive handoffs. A temporal concern graph links symptom, interventions, pending investigations, medication-related issues, deterioration indicators, nursing actions, escalation events, and documented resolutions, enabling the algorithm to distinguish newly identified concerns from continuing, escalated, resolved, and potentially lost concerns. The proposed system is evaluated against conventional structured handoff and checklist-based approaches and computational benchmarks including TF-IDF Support Vector Machine, XGBoost, BiLSTM-Attention, and ClinicalBERT. Comparative performance is assessed using sensitivity, specificity, precision, macro-F1 score, AUROC, area under the precision-recall curve, false-negative rate, calibration error, escalation-detection accuracy, concern-closure accuracy, and processing latency. Additional ablation experiments examine the contribution of temporal memory, escalation weighting, concern-graph modelling, and closure verification. Performance graphs include algorithm-level ROC and precision-recall curves, sensitivity and F1 comparisons, calibration plots, unresolved-concern detection rates, escalation reliability analysis, and component-ablation charts. The central hypothesis is that integrating longitudinal handoff context with explicit escalation and closure reasoning will provide more reliable detection of clinically significant unresolved concerns than isolated text classification or conventional structured handoff methods. HandoffCARE-Net therefore provides a technical foundation for AI-assisted nursing handoff surveillance capable of strengthening information continuity, supporting timely escalation, and improving verification that patient-care concerns are followed through to documented resolution.