You must be able to meet the basic rules or an exclusion. Unemployment is one of the acceptable exclusions through an unforeseeable event (shown below). This will allow, at least, partial exclusion.
You meet the standard requirements if any of the following happened during the time you owned and lived in the home you sold.
Your home was destroyed or condemned.
Your home suffered a casualty loss because of a natural or man-made disaster or an act of terrorism. (It doesn’t matter whether the loss is deductible on your tax return.)
You, your spouse, a co-owner of the home, or anyone else for whom the home was his or her residence:
Became divorced or legally separated;
Gave birth to two or more children from the same pregnancy;
Became eligible for unemployment compensation;
Became unable, because of a change in employment status, to pay basic living expenses for the household (including expenses for food, clothing, housing, medication, transportation, taxes, court-ordered payments, and expenses reasonably necessary for making an income).
An event is determined to be an unforeseeable event in IRS published guidance.
TurboTax will walk you through the process.
I have a related question:
You have a good case, however you mentioned that you quit your job which could possibly be a factor. Regardless, you are only a little over 3 months out from being eligible for the entire exclusion amount so it would be prudent to try and close after that date.