You didn't lay out the details of your case, but typically, the severance package depends on you signing a document in which you agree to certain things, such as, the dismissal is not for any unjust reason (race/religion/gender), and you will not pursue any legal action against the University. Other conditions might be attached as well, such as agreeing not to claim Unemployment Benefits, not to write any bad reviews of employment, not to discuss the details of your severance package, etc.
The severance package doesn't come for free. The employer is typically trading the severance package for the legal certainty of a clean break, and/or the protection of their reputation.
If you decline to sign the document, and do not accept the severance package, the implication is that you're not releasing the University, and you might sue, or might claim benefits, or might write bad reviews, etc.
Your employer would weigh your suggestion of a bigger/better benefits package against that. Do they think you could reasonably sue them for anything? Do they really care if you write some bad reviews? Does it affect them much if you claim a few weeks of Unemployment Benefits?
If they answer these questions as "No, not likely, and if it does happen, its not a big deal", then they'll probably let you go with no severance package at all.
If you think you really can concern them with a legitimate issue (eg. "I'm not willing to sign this package, because I don't think the office-bullying has been properly addressed"), then maybe they'll offer something more.
Be extremely careful not to phrase it in such a way that sounds like blackmail, such as:
"If you don't give me a bigger package, I just might talk to a reporter about the sexist workplace culture". That would be crossing a line that might expose you to criminal charges.