The Belmont Club continues to produce good stuff. It has a few good pages on what may be going on in Fallujah, with a footnote on Najaf. The short of it is this:

The enemy is probably still in the city
the enemy may consist, in part, of Syrian fighters;
the USMC is probably still bottling them up otherwise how to account for the enemy containment, and is therefore present in the city, contrary to press reports;
the USMC is attempting to drive a wedge, as per General Conway, between the hard core and the peripheral enemy elements. Thoughtful, read it all.