Seems that the net is not exactly full of datasheets of these transformers.
You can start checking how secondary side reacts when load increases.
All energy goes trough that transformer, so it can be as it should.
TL431 is referencing how PS2561 is driving the switcher.
Transformer's secondary side before diode pair should have a clear effect when load changes.
Output voltage should also be that 12V and not follow the load.