I've given it some thought...And I realized I never got an offer letter for the current job I have. It was all over the phone until I got the contact to sign, so there's a good chance there's nothing to worry about. Except it'll give your parents fuel for the fire.
Does your current job know you've been searching for a job? If so, you could let them know and have the notice for when you pass the drug test. Aside from that, just prep as much as you can for the move.
This might be just one of those cases in life where you just have to take a deep breath and go for the plunge without total certainty.