Speed/Effort:
* Slowly: "He dragged the heavy box slowly across the room."
* Laboriously: "She dragged the fallen tree branch laboriously back to the woods."
* Wearily: "The hikers dragged themselves wearily towards the campsite."
* Reluctantly: "He reluctantly dragged himself out of bed."
Direction/Force:
* Awkwardly: "He awkwardly dragged the suitcase through the door."
* Violently: "The storm violently dragged the boat out to sea."
* Forcefully: "She forcefully dragged the child away from the edge."
* Carefully: "He carefully dragged the injured bird to safety."
You could also use adverbs that describe the feeling of the action:
* Painfully: "He painfully dragged his broken leg to the hospital."
* Frustratingly: "The car frustratingly dragged its way up the hill."
Ultimately, the best adverb to use depends on the context and the specific meaning you want to convey.