Healthcare workers who simultaneously provide patient care and manage telecare sensor monitoring systems represent a vulnerable workforce balancing caregiving and technical responsibilities. This study examines security vulnerabilities in the LightAuth lightweight authentication framework deployed in smart healthcare sensor networks used by these dual-role workers. Analysis reveals critical weaknesses: insufficient freshness verification enabling replay attacks with 82% baseline detection, flawed key agreement susceptible to man-in-the-middle manipulation, and persistent exposure of static patient/worker identifiers undermining privacy. We propose practical countermeasures including timestamp+nonce validation, mutual signature verification, dynamic pseudonyms, and perfect forward secrecy integration. Experimental evaluation on Raspberry Pi sensor nodes demonstrates that enhanced protocol achieves 100% replay detection while incurring acceptable overhead: authentication latency increases from 38.5ms to 46.8ms (21%), and throughput reduces from 290 to 245 TPS (15%), remaining within practical limits for telecare workloads supporting dual-role healthcare workers.