First off, I absolutely disagree with not contacting them if it's been more than a few weeks. They are a small staff that are very busy. I have anecdotes but, trust me. Send a nice email to Angie or whoever gave you their card when you interviewed, and I promise it won't cost you your job. They'll likely thank you for contacting them.

They do need the drug screen before you can go to class. They send out the requests once they give you a conditional offer. Your current airline has 30 days to respond.