Do I need to worry about my background check affecting me negatively if I had a criminal conviction, count for possession, filed in 2003?

I expunged this later when I was allowed to. I had a DWI at a later date, approximately 2005/2006 (honestly don’t remember when exactly). I am now living in another state and applied for a job. They use a third party background investigation service who claims they searched back by two most previous address. Being it is 2017, my application didn’t state any past criminal convictions because I was told in court by the judge not to worry and that there is no need to admit charges on any job applications. This applies to my DWI at the later date as well because it has been about 10 years already. I verbally informed the employment office of my court direction regarding being able to apply and not state any past convictions. I am stressed and worried because I have worked really hard in my life and completely turned around a long time ago.