One thing to make sure which you probably did is that you need the instance to be General Purpose or Memory Optimized SKU.
In the blade of the PostgreSQL instance you should have Logs option on the left menu. That logs option allows you to query Log Analytics directly and any query you enter will automatically be scoped only to the logs that are generated by the PostgreSQL instance of the blade.
With diagnostic logs you have to keep in mind that these become available in AzureDiagnostics table only when such log is generated. If such log is not generated you might see the table empty for your PostrgeSQL instance until log is generated. In case you are logging only errors and warnings you might want to set the minimum log level on the PostgreSQL instance so you generate more log levels.