Am I eligible for severance pay from university? How?

Contact an attorney to review your matter and determine your rights.

You did not identify your state, the answer varies from state to state. If you are in Georgia: There is no law that requires any employer to pay severance pay. Such matters are governed by company (University) policy, and perhaps by your employment contract, if you have a contract. Contracts are unusual, you likely have none. Under GA Law, you are entitled to no severance pay. You likely qualify for unemployment comp benefits, $330 a week for 19 weeks, no extensions. Back to University policy. Go to your HR department, or a trusted manager, and ask. Universities are usually open to discussion about such matters. You may be surprised at what they offer. Once you have an offer, you may want to discuss this with an attorney. Sometimes attorneys can negotiate a bit more. Other times attorneys kill the deal.

Depends on your contract and the employment practices of the university.