Temporal graph neural networks (TGNNs) have shown remarkable performance in temporal graph modeling. However, real-world temporal graphs often possess rich textual information, giving rise to temporal text-attributed graphs (TTAGs). Such combination of dynamic text semantics and evolving graph structures introduces heightened complexity. Existing TGNNs embed texts statically and rely heavily on encoding mechanisms that biasedly prioritize structural information, overlooking the temporal evolution of text semantics and the essential interplay between semantics and structures for synergistic reinforcement. To tackle these issues, we present \textbf{CROSS}, a flexible framework that seamlessly extends existing TGNNs for TTAG modeling. CROSS is designed by decomposing the TTAG modeling process into two phases: (i) temporal semantics extraction; and (ii) semantic-structural information unification. The key idea is to advance the large language models (LLMs) to dynamically extract the temporal semantics in text space and then generate cohesive representations unifying both semantics and structures. Specifically, we propose a Temporal Semantics Extractor in the CROSS framework, which empowers LLMs to offer the temporal semantic understanding of node's evolving contexts of textual neighborhoods, facilitating semantic dynamics. Subsequently, we introduce the Semantic-structural Co-encoder, which collaborates with the above Extractor for synthesizing illuminating representations by jointly considering both semantic and structural information while encouraging their mutual reinforcement. Extensive experiments show that CROSS achieves state-of-the-art results on four public datasets and one industrial dataset, with 24.7% absolute MRR gain on average in temporal link prediction and 3.7% AUC gain in node classification of industrial application.