Shy Introvert, speaking as someone who has done the hiring before, it sounds like you are "hired pending background check." Which means that they could still rescind the offer and that it's not entirely yours yet until you pass the background check and drug test and have an official start date. It does sound like everything is lining up though. I don't think the valium will affect your hiring. If they choose not to hire you, they should tell you if it's because of a background check or failed drug test. If it's a prescribed medication, it shouldn't cause you to fail the drug test.

I would keep looking while you wait for an official start date (just to be safe), but it does sound fairly solid, pending background check and drug test. All of my hirings have been made as offers pending background checks, and we have to do our due diligence. As long as there is nothing on your background check, sounds like you will have a start date soon.

Congrats!
Seesaw