An educational technology startup has developed a serverless learning platform using AWS. The platform leverages Amazon API Gateway, AWS Lambda, and an Amazon RDS for PostgreSQL instance to manage its backend. Recently, the development team observed a spike in connection timeout errors between the Lambda functions and the database, especially during periods of erratic or high user activity. The team wants to reduce these failures while minimizing the effort and changes needed to the existing codebase. What is the most effective approach to address this issue?