*****
 My first diagnosis was that I was queenless in the hive even
     though the hive was behaving normally.  It wasn't overly
     aggressive (even after I dropped one of the brood chambers),
     bees were flying and taking advantage of the weather and they
     had put up 4 full medium supers of honey.
******
What you describe is similar to what I often see in a hive a few weeks after
it swarms.  The last offspring from the old queen have emerged, but the new
queen (assuming one is there) has not yet begun to lay.  If the bees are
behaving normally, then they likely have a virgin or a newly-mated queen.
 
This period should only last a week or so.  If you don't have eggs by then,
you probably don't have a queen.
 
Opened swarms cells would tell you whether swarming had occurred.