They won't know you are selling your house unless you tell them. Whether having the mortgage makes a difference or not is dependent on your overall financial position and not something we could or should guess at here.
The info about the house sale may be hard to work around these days. A computer will evaluate your application and it will decide/recommend based on the info it has. Until the mortgage is actually gone, it appears as "debt" and definitely has an impact. There may be a way to tell the computer to ignore the info, but you would have to work that out in advance. You might call the GS loan agent ahead of time and ask how best to handle that. Also, even a negative computer recommendation may get a human review - that's a matter of procedure at whatever lender GS is using. If there is a review and they contact you about it, you could them introduce the house sale info.
War story: I ran into a computer processing glitch several years ago at BOA. I wanted to refinance my existing BOA RV loan and their procedure was for me to submit an RV loan app to start the process. I did that, but the computer rejected it because the RV was "too old" - they weren't giving loans for RV's over age 5 at the time. I called the BOA agent back but there was no application in his computer system for him to review or override. There was no way in their computerized application to tell it this was a re-finance of an existing loan rather than a new one, and no way for them to apply human judgement without an application in the computer system! Bizarre! The only way in their system to handle it was if I was trading in the current RV on a new one. I gave up on them after a couple weeks of excuses because nobody at the working level in BOA's RV loan division knew a way around the processing procedure problem.
