I think you are good to go. If they were going to make your offer contingent on passing the background screens they would also have made it contingent on passing a drug screen. They would want the results before or very close to your start date and not have to pay you and process paperwork for a couple of days of work. Just read your info and employee handbook carefully and look for any mentions of drug tests. Some employers will require a drug test if you are invloved in an accident at work. They may cover this in orientation. Good Luck.