Sepsis and septic shock are a critical medical condition affecting millions globally, with a substantial mortality rate. This paper uses state-of-the-art deep learning (DL) architectures to introduce a multi-step forecasting system to predict vital signs indicative of septic shock progression in Intensive Care Units (ICUs). Our approach utilizes a short window of historical vital sign data to forecast future physiological conditions. We introduce a DL-based vital sign forecasting system that predicts up to 3 hours of future vital signs from 6 hours of past data. We further adopt the DILATE loss function to capture better the shape and temporal dynamics of vital signs, which are critical for clinical decision-making. We compare three DL models, N-BEATS, N-HiTS, and Temporal Fusion Transformer (TFT), using the publicly available eICU Collaborative Research Database (eICU-CRD), highlighting their forecasting capabilities in a critical care setting. We evaluate the performance of our models using mean squared error (MSE) and dynamic time warping (DTW) metrics. Our findings show that while TFT excels in capturing overall trends, N-HiTS is superior in retaining short-term fluctuations within a predefined range. This paper demonstrates the potential of deep learning in transforming the monitoring systems in ICUs, potentially leading to significant improvements in patient care and outcomes by accurately forecasting vital signs to assist healthcare providers in detecting early signs of physiological instability and anticipating septic shock.