Auto-bidding is a crucial task in real-time advertising markets, where policies must optimize long-horizon value under delivery constraints (e.g., budget and CPA). Existing methods for auto-bidding rely on compact numerical state representations: while they can implicitly capture delivery dynamics, they offer limited support for explicitly representing and controlling high-level intent, evolving feedback, and operator-style strategic guidance in real campaigns. Meanwhile, Large Language Models (LLMs) offer a powerful method for encoding semantic information, it remains unclear when LLMs help and how to integrate them without sacrificing numerical precision. Through systematic preliminary studies, we find that (1) LLM embeddings contain bidding-relevant cues yet cannot replace numerical features, and (2) gains emerge only with careful semantic--numeric integration rather than naive concatenation. Motivated by these findings, we propose \textit{SemBid}, a novel auto-bidding framework that injects LLM-encoded semantics into offline bidding trajectories at the token level. SemBid introduces three semantic inputs: \textit{Task}, \textit{History}, and \textit{Strategy}. It injects these semantics as tokens alongside numerical trajectory tokens and uses self-attention to integrate them, improving controllability and generalization across objectives. Across diverse scenarios and budget regimes, SemBid outperforms competitive baselines from offline RL and generative sequence modeling, with more consistent gains in overall performance, constraint satisfaction, and robustness. Our code is available at: \href{https://github.com/AlanYu04/SemBid-KDD2026}{\textcolor{blue}{here}}.