The saturation rating refers to the algebraic total current, net curent, through the transformer. Current in the secondary adds, preserving sign, to current in the primary.
For convenience, assume 1:1 windings. For a transformer with a different ratio 1:N, just multiply the appropraite winding current by N.
When you apply the input voltage waveform with no output load, so zero secondary current, the primary current should stay below the satauration limit, if you have designed it with enough primary turns. Any given core will have a volts per turn at a given frequency, and you need enough turns to support the input voltage.
When you connect an output load, a secondary current will flow, which increases the primary current, such that the net current stays pretty much as before (exactly the same as before in a transformer with no primary resistance). The secondary current cancels out the primary current increase.
As the input voltage is the same, the net current through the coils is the same, as it's changes in this net current that creates the back emf that balances the primary input voltage.
In a transformer with primary resistance, the extra current through the primary creates an IR voltage drop, which subtracts from the input voltage, so the net current falls slightly under load, as it is supporting a slightly smaller input voltage across the primary inductance.