You can drag him by holding E. I ended up telling him I was there to kill him in his room, cut him down as he ran out, and then dragged him behind the wine barrels and that counted.
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Acquiring the Bonus
To acquire the bonus, Balagog's body needs to be hidden somewhere. Back in the Sanctuary, Gabriella hinted that she uses bodies of water for hiding victims. The innkeeper of the Nightgate Inn, Hadring, mentions that Balagog often goes down to the lake. So, the Dragonborn can wait until he leaves (around 12:00 p.m. to 1:00 p.m.), then follow him to the lake, kill him, and drag his body into the water (the nearby river will not count). Due to his low health, even attacks such as Unrelenting Force will kill him instantly. A high damage ranged attack may throw him in the water. At that point, his body can be looted, which will complete both tasks.
If he is killed in the cellar of the Nightgate Inn, there are several ways to hide his body. One is to drag his body and lift it into the large empty wine barrel at the east end of the room. Removing everything from his inventory will make his body lighter, and may be necessary to lift him into the barrel. Another way is to drag him behind the hay bales until this part of the quest is marked as completed. Dragging the body under the bed also works. Finally, if he is spoken to and told that he is going to be killed, he will run and try to hide behind a vat - if he is then killed there, his body is hidden enough to satisfy the "hide his body" objective.
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If you’re having trouble figuring out how, that’s because there really isn’t a prompt for it, it just says “Press such and such button to loot”, but if you press and HOLD the button you normally press to loot the body, you will grab the body and drag it with you as you move.
The best way to move a body isn’t by dragging, but by grabbing the head or ankles and picking it up off of the ground completely to move it. When trying to drag a body, the ground creates a lot of friction, so you’ll usually end up letting go of the body multiple times before you’re done.
Moving other objects around is the same, simply press and hold the button you would normally press to TAKE the object to pick it up and move it. This is not considered stealing either.