Does a saturated inductor radiate more?
Once the core is saturated, it doesn't look like a magnetic core anymore when you try to increate the magnetic field.  The inductor becomes essentially air-core for any additional current.

While over-saturating a shielded inductor will cause the magnetic field to leak outside the shield more, this shouldn't cause much radiation by itself.  More likely the sudden decrease in inductance causes large current spikes somewhere.  These can cause radiation, depending on how they are routed, and therefore how much they act as antennas.