Received job offer today, but quit current job two weeks ago - How to proceed?

When I applied for this job and interviewed, I was still employed. But then two weeks ago, I quit. Today I received a job offer. I was only employed at the previous company for a month.

Now I have to fill out the background check tomorrow and I don't know how to proceed. Should I just continue along as if I'm still employed there? I know the backcheck is completed by Sterling and I think if I put "Do not contact" for "current employer" then they will just ask for a recent paystub and won't go digging around for further details.

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