You would not have the grappled condition as the spell does not mention the grappled condition. As was previously stated(though maybe not anymore), mimics have a similar ability and specifically call out grappled as part of their attack. This gives precedence which supports no grapple. You could, however, both now effectively be tied together though. How that jives together is up to house rule I guess.
Exactly what happens if you hold on to your weapon, especially if the weapon is attached to you, and try to run is much less clear.
The combat maneuver drag clearly states that: "The aim of this maneuver is to drag a foe in a straight line behind you without doing any harm." If that is not your aim, then you are, by definition, not attempting to perform a drag maneuver. It would also require you to make an attack in order to attach yourself somehow to your opponent, something you have no need to accomplish. Of further note, the drag combat maneuver is not in the CRB, so may not even be an option on the table.
Basically this spell leaves you in doozy of a crap rules situation. The easiest and quickest solution is to see what your DM has to say on the situation.