This happened to me in 2016/17. I lost my job 12/31/16, thankfully found a new job starting early January 2017. But my severance was paid 1/2017. Since we were pretty much full pay, it only made us lose the subsidized portion of the federal loan. I didn’t bother to appeal because I did find a job immediately, but the school does have an income appeal process.
Something else to be aware, not sure if it will impact you - I rolled my 401k over when I started my new job. If you do the online retrieval with the IRS, they will count that money as income. I knew the severance would hurt, but when our EFC went from about $25k to $175k, I knew something was off! I had to work with the university directly on that too.