A common misconception regarding transformers is that overloading a transformer will cause it to saturate. More technically, this implies increasing the load level increases the flux density within the core. This isn't true, under a more nuanced lens the opposite actually occurs; increasing the load level of a transformer decreases the flux density within the core.
Lets construct a transformer equivalent circuit from real-world values from the Practical Transformer Characterization Article:
Core-Loss Resistance
Magnetizing Impedance
Combined Winding Resistance
Combined Leakage Reactance
We can now subject the transformer to three levels of loading:
Since the induced voltage in any coil is proportional to the flux density we can then say:
Clearly, the loading of a transformer and the operating / peak core flux density exhibit an inverse relationship, which is the opposite of what many expect.