LLM-driven sequential recommendation formulates next-item prediction as autoregressive generation conditioned on natural-language prompts. However, minor wording changes in semantically equivalent prompts can cause substantial performance fluctuations, undermining robustness and requiring costly manual prompt engineering. Continuous prompt learning reduces template dependence but faces two interacting challenges: shared task-level instructions lack user-specific reasoning guidance, while gradient updates can push continuous prompts outside the LLM's effective semantic space. Injecting personalized signals can further amplify this semantic drift. To address these challenges, we propose LRPRec, a learnable prompting framework that initializes continuous instruction prompts from discrete templates and introduces two complementary mechanisms. Personalized prompt injection encodes user behavior into a preference embedding and additively injects it into shared prompts, enabling parameter-efficient user-level adaptation. A semantic drift constraint regularizes the shared prompts within a trust region around their initialization anchors to preserve semantic validity during optimization. By constraining the shared component while allowing additive personalization, LRPRec decouples stability from expressiveness. Extensive experiments on three benchmark datasets demonstrate consistent improvements over strong baselines while eliminating the need for manual tuning of background and task inference templates.

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