Foundation models (FMs) are large-scale deep neural networks trained on vast and diverse datasets, capable of learning transferable, domain-agnostic representations that enhance performance across a wide range of downstream tasks. These models have attracted significant attention across multiple fields, potentially reshaping standard paradigms of model design. However, adapting FMs to specific tasks typically requires computationally intensive fine-tuning and large labeled datasets, which can be significant limiting factors, especially in resource-constrained settings. In this work, we explore an alternative strategy: leveraging pre-trained FM embeddings (also called vector embeddings) as inputs to downstream supervised models. By utilizing these rich, general-purpose representations, the approach retains the expressive power of FMs while significantly reducing both computational costs and data requirements. We present a general framework, namely CURE-FM (Context-aware Use of data REpresentations with Foundation Models), designed to extract embeddings from self-supervised FMs trained on raw data, and to leverage downstream models for generic tasks. We present an application of CURE-FM relying on raw time series data; in particular, the application is geared towards binary classification tasks over multi-lead electrocardiogram (ECG) signals. We report a comprehensive evaluation, including ablation studies on FM and classifier architectures, structural and explainability analyses of the embeddings, and a comparison between our approach and common fine-tuned based approaches, with the aim of investigating how different architecture and approaches affect performance and generalizability. Experimental results show that embedding-based approaches can provide a scalable, robust, and efficient solution for downstream tasks, and that CURE-FM holds significant promise for advancing ECG analysis.