Latent Diffusion Models (LDMs) have established themselves as powerful tools in the rapidly evolving field of image generation, capable of producing highly realistic images. However, their widespread adoption raises critical concerns about copyright infringement and the misuse of generated content. Watermarking techniques have emerged as a promising solution, enabling copyright identification and misuse tracing through imperceptible markers embedded in generated images. Among these, latent-based watermarking techniques are particularly promising, as they embed watermarks directly into the latent noise without altering the underlying LDM architecture. In this work, we demonstrate that such latent-based watermarks are practically vulnerable to detection and compromise through systematic analysis of output images' statistical patterns for the first time. To counter this, we propose SWA-LDM (Stealthy Watermark for LDM), a lightweight framework that enhances stealth by dynamically randomizing the embedded watermarks using the Gaussian-distributed latent noise inherent to diffusion models. By embedding unique, pattern-free signatures per image, SWA-LDM eliminates detectable artifacts while preserving image quality and extraction robustness. Experiments demonstrate an average of 20% improvement in stealth over state-of-the-art methods, enabling secure deployment of watermarked generative AI in real-world applications.