Coupled Inductor/Magnetic Circuit Behavior
Is this correct?

Yes, theoretically that is correct. As soon as the power supply starts to limit current, it's the end of the output voltage being constant. This is because induction stops i.e. the flux is no longer changing linearly with time because the current has abruptly halted at some value.
Given that voltage induction into the secondary is proportional to rate of change of primary magnetization current, the moment the primary current stops changing, the secondary voltage exponentially decays to zero based on the secondary inductance and the value of the resistive load. This phase is shedding the energy from the magnetic field. Here's a simulation: -

1 volt DC is applied at t = 0 and the following plot shows the secondary voltage (in red) and the primary current (in green): -


How does this affect conditions in the primary?

At the point when the current limit occurs, the primary current remains fixed and the primary voltage follows the secondary voltage (1:1 turns ratio) and decays to zero volts.

What is happening with the fields from each winding?

There are three fields: -

magnetization field
field caused by secondary current flow into the load
field caused by secondary current flow into the load but from the primary

Fields 2 and 3 always cancel out and field 1 remains: -

Picture from here.

What energy does each hold?

The only energy is held in the magnetization field, \$½\cdot L\cdot i^2\$.