Well, kind of. You'd have to bring the difference with you to the closing. You could get a personal loan for the difference, prior to the sale, and then show up with the cash. Recognize, though, that it'll be a lot more than $6,000. If you use a real estate agent, there will be commissions. The buyers may require you to help pay their closing costs. There might be repairs needed to the house. Allow, at a minimum, another $6,000-$8,000. So, with your scenario, you'd have to borrow $12,000 or more and then pay that out at closing. It certainly can be done.
You can check with your Realtor for a better estimate of the expenses you're likely to incur.
Hope that helps.