Navigating human-filled spaces is crucial for the interactive social robots to support advanced services, such as cooperative carrying, which enables service provision in complex and crowded environments while adapting behavior based on real-time human language commands or feedback. However, existing social robot navigation planners face two major challenges: managing real-time user inputs and ensuring socially compliant behaviors in unfamiliar, zero-shot environments. In response, we introduce SALM, an interactive, human-in-loop Socially-Aware navigation Large Language Model framework that dynamically integrates deep reinforcement learning (DRL) with large language model (LLM) capabilities. SALM leverages contextual semantic understanding from real-time human-robot interactions to convert high-level user commands into precise, low-level control actions. A high-level LLM module parses user input, guiding the simultaneous generation of navigation commands by both a large language navigation model (LNM) and a DRL-based navigation model (RLNM). A memory mechanism archives temporal data for continuous refinement, while a multi-step graph-of-thoughts inference-based large language feedback model adaptively fuses the strengths of both planning approaches. Experimental evaluations demonstrate that SALM not only enhances navigational precision in crowded, dynamic environments but also significantly improves system adaptability, offering tailored behaviors that align with individual user preferences and real-time feedback. More details and videos about this work are available at: https://sites.google.com/view/navi-salm.