Social media platforms such as Twitter (now X) provide rich data for analyzing public discourse, especially during crises such as the COVID-19 pandemic. However, the brevity, informality, and noise of social media short texts often hinder the effectiveness of traditional topic modeling, producing incoherent or redundant topics that are often difficult to interpret. To address these challenges, we have developed \emph{TM-Rephrase}, a model-agnostic framework that leverages large language models (LLMs) to rephrase raw tweets into more standardized and formal language prior to topic modeling. Using a dataset of 25,027 COVID-19-related Twitter posts, we investigate the effects of two rephrasing strategies, general- and colloquial-to-formal-rephrasing, on multiple topic modeling methods. Results demonstrate that \emph{TM-Rephrase} improves three metrics measuring topic modeling performance (i.e., topic coherence, topic uniqueness, and topic diversity) while reducing topic redundancy of most topic modeling algorithms, with the colloquial-to-formal strategy yielding the greatest performance gains and especially for the Latent Dirichlet Allocation (LDA) algorithm. This study contributes to a model-agnostic approach to enhancing topic modeling in public health related social media analysis, with broad implications for improved understanding of public discourse in health crisis as well as other important domains.