Refactor DbContextManager to correctly handle async sessions and increase pool size

[edulauer]

Fix #229
This PR refactors the DbContextManager to follow async SQLAlchemy best practices and to avoid QueuePool limit error by:

• Storing the async_sessionmaker factory instead of creating a session in init.
• Moving session creation to aenter so each context gets a fresh AsyncSession.
• Ensuring proper commit/rollback handling in aexit.
• Increase pool_size and max_overflow limit (from 5 to 15)