No, don't close your oldest card or any card before making a mortgage application. Don't make changes to your CR before you get your mortgage. Wait until you close on your home and then close the credit card if you still want to close it. Making changes before applying for a mortgage is not a good thing - except for paying down debt. Critical Point: you want your debt paid and reporting the new balance to the CRA's before you apply for your mortgage, not after you have made your mortgage application.
Pay it off as you plan to do so. That way your new zero balance can help your score (zero utility is a good thing). As elsewhere on this site, if you can have all of your revolving accounts paid to zero except one credit card reporting a balance of 1% to 9%, that will optimize your score.
By the way, you AAoA won't drop in FICO scoring closing the oldest account as it continues to report for up to 10 years after you close it. [However, closing it affects your AAoA for Vantage scores immediately.] But closing the account will affect your utilization and that will have a direct impact on your FICO score. Losing utility usually hurts your score, so don't risk it before you close on your mortgage
