Digital Twins (DT) and Extended Reality (XR) technologies are leading the way in industry transformation by bridging the real and virtual worlds and enabling real-time user-digital system interaction and simulation. Despite their promise, XR-enabled DTs still face considerable difficulties in ensuring smooth synchronization between the actual and virtual worlds, especially in network-constrained scenarios. This paper addresses these challenges by introducing aDapT-XR (Adaptive Data Allocation and Prioritization Technique for DTs in eXtended Reality), a novel network-aware data prioritization framework designed for heterogeneous data flows in real-time XR environments. aDapT-XR dynamically adjusts data transmission priorities according to network conditions, reducing latency and improving quality of service (QoS) levels. This work addresses the main limitations of existing methodologies and contributes to the advancement of XR-DT system design by offering practical solutions for live shows, industrial monitoring, and remote collaboration.