IA = (VT - EA)/X. As the phase angle of the back emf in the armature EA lags the phase angle of VT by a larger angle δ when the rotor slows due to a mechanical load increase, the numerator of the equation goes up, and therefore so does the armature current. Since torque is proportional to armature current, it also increases as the angle increases (which is why I mentioned the angle is called the torque angle) unless maximum torque is exceeded and pole slip occurs when the magnetic linkage breaks as you state. Note that the magnitude of the armature voltage EA only changes if the excitation changes, so unless that occurs, only the torque angle increasing is responsible for the increase in armature current when the load increases.
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