If the overdraft was paid by your bank and no payments were late to creditors, it could not affect your score. If the overdraft was not paid by the bank and, for example, a credit card payment became not paid on time, then it will eventually find its way into your score. Typically, it takes a few weeks to appear.
So, if you are anticipating closing within a couple of weeks, it is very unlikely your score will be affected.
Your bank balance has been affected, though. If you claimed that you had $3,000 in the bank, for example, and now it is bouncing around zero, the lender may discover this when he verifies deposits and then will inquire where you will get funds to close with.
Lenders have the right to pull your score at any time. Typically, it is pulled by the loan officer, then later by processing, and may be pulled the day of closing or the day before at the behest of underwriting.