fix(dbmanager): keep the pool alive across errors and restarts

Implements the fixes from audit/pkg/dbmanager.audit.md.

- Stop closing the shared *sql.DB to recover from errors. Adapter
  factories and the health checker no longer call Reconnect; Reconnect is
  atomic and operator-only.
- Postgres uses a custom driver.Connector: Reconnect retires pooled
  connections by generation without closing the pool, so held Bun/GORM
  handles keep working. Verified against a live server restart.
- Add TCP keepalive, TCP_USER_TIMEOUT, a bounded reuse ping and
  statement_timeout as a runtime parameter; drop the 2 min timeout floor.
- Health check pings without holding the connection lock.
- Listener: single goroutine pair, bounded Close without UNLISTEN, and
  serialised use of the pgx connection (fixes conn busy and a close race).
- Fix Connect/Close/Connect/Close panic, idempotent Connect, dial outside
  the manager lock, clean up on partial failure.
- SQLite: pin :memory: to one connection, pragmas via DSN.
- Escape credentials in Postgres/MSSQL/Mongo DSNs; sslmode defaults to
  prefer. Wire retry settings, publish metrics, fix logger calls.
- NewConnectionFromDB: Close is a no-op with a warning (caller owns the
  pool); Reconnect only pings.
- Document correct usage in the README; mark the audit with what was done.

Co-Authored-By: Claude Sonnet 5.5 <noreply@anthropic.com>

pkg/dbmanager/providers/existing_db_test.go

@@ -164,7 +164,7 @@ func TestExistingDBProvider_Stats(t *testing.T) {
 	}
 }
 
-func TestExistingDBProvider_Close(t *testing.T) {
+func TestExistingDBProvider_Close_LeavesDBOpen(t *testing.T) {
 	db, err := sql.Open("sqlite3", ":memory:")
 	if err != nil {
 		t.Fatalf("Failed to open database: %v", err)
@@ -177,10 +177,10 @@ func TestExistingDBProvider_Close(t *testing.T) {
 		t.Errorf("Expected Close to succeed, got error: %v", err)
 	}
 
-	// Verify the database is closed
-	err = db.Ping()
-	if err == nil {
-		t.Error("Expected database to be closed")
+	// The caller owns the database, so Close must leave it open
+	defer db.Close()
+	if err := db.Ping(); err != nil {
+		t.Errorf("Expected caller's database to stay open, got: %v", err)
 	}
 }