Audio is a rich sensing modality that is useful for a variety of human activity recognition tasks. However, the ubiquitous nature of smartphones and smart speakers with always-on microphones has led to numerous privacy concerns and a lack of trust in deploying these audio-based sensing systems. This paper addresses this critical challenge of preserving user privacy when using audio for sensing applications while maintaining utility. While prior work focuses primarily on protecting recoverable speech content, we show that sensitive speaker-specific attributes such as age and gender can still be inferred after masking speech and propose a comprehensive privacy evaluation framework to assess this speaker attribute leakage. We design and implement FeatureSense, an open-source library that provides a set of generalizable privacy-aware audio features that can be used for wide range of sensing applications. We present an adaptive task-specific feature selection algorithm that optimizes the privacy-utility-cost trade-off based on the application requirements. Through our extensive evaluation, we demonstrate the high utility of FeatureSense across a diverse set of sensing tasks. Our system outperforms existing privacy techniques by 60.6% in preserving user-specific privacy. This work provides a foundational framework for ensuring trust in audio sensing by enabling effective privacy-aware audio classification systems.