Col. Bob Garner said authorities believe the murder happened "in the heat of the moment" without premeditation.
The victim, Henry Ramirez - Crawford's grandfather - was killed on Jan. 6, records state. Police said that Ramirez was killed after he refused to give Crawford and Cole, her boyfriend, money.
Ramirez's body was found wrapped in blankets and garbage bags and buried off Jenkins Road on Jan. 11.
Garner alleged that Cole and Crawford drove Ramirez's body to West Virginia in the back of a Budget rental truck. Once here, they asked to borrow a shovel and wheelbarrow from someone who lived near where the body was found, he said.
After the items were returned, the person who had loaned them went to the burial site and found the body, Garner said.
Both Cole and Crawford will continue to be held in Eastern Regional Jail without bail, Berkeley County Prosecutor Pamela Games-Neely said.
Suspects wanted in another state can fight going back if they maintain they are not the people named in the paperwork, were not in the state at the time of the crime or if they believe the other state's paperwork is not in order. They cannot fight extradition because they believe they are innocent and want to fight the allegations, Games-Neely said.
Because Cole and Crawford said they are not going back voluntarily, the governor of Louisiana will need to issue a governor's warrant demanding their return, Games-Neely said.
A murder charge filed in Berkeley County against Crawford will be dismissed, Games-Neely said. Cole was not charged in West Virginia in connection with the death, but is being held in Eastern Regional Jail on unrelated fugitive warrants from Virginia.
Games-Neely said she does not know whether Virginia authorities will dismiss their charges.