Inductance changes in DC relay coils
I was unable to reproduce this behavior.

I found a relay with a removable cover (stolen from my heating system!) and connected an LCR meter to the coil. 

I tested the inductance of the coil with the relay open and found it to be 360 uH.

Then I manually closed the relay arm by pushing it with my finger and found the inductance to go up to 585uH.

This is what I'd expect to see since moving the arm closer to the core of the relay's coil should increase the flux though it, thus increasing the total inductance of the coil. 

So, possibly...


The inductance change you are seeing is really an artifact of your PWM strategy. 
Your relay is very different than mine. 


I'd recommend you try try testing the inductance with the relay open and manually closed (not energized in either test). This will give hints as to where to look next. If you again see the inductance unexpectedly go down, then we have something interesting here!