Connection pooling at scale: managing 10K+ database connections

we're growing fast and our microservice count is now over 200, each maintaining its own database connection pool. we're frequently hitting postgresql connection limits, even with moderately sized pools. this is leading to connection timeouts and service instability. we've started looking at pgbouncer as a centralized connection pooler to multiplex connections, but it's a significant operational shift. we're also considering whether reducing application-level pool sizes and relying more on pgbouncer is the way to go, or if we should even consider migrating some high-volume, simpler services to a nosql database like dynamodb to offload postgresql. what strategies have proven effective for managing thousands of database connections across hundreds of microservices without constantly battling connection limits?