Comcast Hiring Process; background check; Need Help!

I'm confused of the hiring process. So I had a face to face interview 4 days ago. The next day, the HR gave me a verbal offer contingent upon background/drug check. I was told to fill out some release form and so I did.
Comcast training starts in 3 weeks. I have a job that requires me two give a 2-weeks notice. If the background check is not cleared this week then I'm in a really bad situation.
I am not too terribly worried about my background and all that but I think I may have missed something in the employment history. I moved here in the US 7 years ago and my first year was a mess. I hopped from job to job and can hardly remember the exact dates of each job. I was adjusting to the new environment and of course everything was new to me. However, I have a solid 5 years of experience in Customer Service, IT and Sales at my current job. Plus I have a degree from overseas. No misdemeanor and I have paperwork to prove that I am a legal to work here in the good ole USA.

If I missed to list one odd job that I had 7 years ago, should I be worried? How long does the background check normally takes? Should I go ahead and give my boss a 2-week notice before I even find out if my background check has cleared?

I am so stressed out thinking about this whole thing and I need someone's help.

I have worked in a Comcast call center for 3 yrs in Customer Service (Billing & Repair). My background check took right at 1 week. I too had trouble remembering exact hire dates going back 5-7 yrs. There were a couple of dates that I had wrong. The HR recruiter called me once the background check was done and ask a few questions. I told the truth...that I put down the dates to the best of my knowlege. I even had one former employer tell the background company that I was not eligible rehire. I still wonder if there was a mistake, as I left this company in very good standing as a layoff (not fired or quit). Anyways, the recruiter was very fair and pretty relaxed about it. After telling her my side of the story she said that she believed me because all other employers had given me good recommendations and she didn't even call the employer that gave me the bad recommendation. I don't think you have anything to worry about. Not everyone can get the dates and all details exactly correct going back that far.

I just got an email from comcast with their position offer letter! Background check went fine. The only problem is, they wanted me to start on 4/25 (today is 4/13). I told them that I have to give a solid 2-weeks notice to my current employer and asked if I could start in May instead (another training starts in May 23). I do hope they understand. I told the other comcast HR about this on my first interview and she probably forgot that I am currently employed.

well they called me this murin ask me when i graduated college..cuz i made a mistake on my app..than she said ok well ill put inn ur grad date N we will contact you about tha drug screening...tis iz such a long process..im goin let god handle it..i jst really want tha benefits