The key to your question is the agreement with the DI insurer and with the State. By buying the DI coverage afrom the DI insurer and accepting money from the State, you promised and they promised to do certain things. Get the agreements and see what they promised and what you promised. It is hard to say what you can do until you get the agreements. They may be entitled to all of it, or none of it.
I wish I could give you a definitive answer, but your state's laws, as well as the policy language itself will determine the answer. If you have a lawyer, the lawyer should be able to answer this question. If not, I would start by asking for a copy of the policy from your employer.
If there is no language requiring repayment, you still need to know if there is a state law requiring the state to be paid back. That is best answered by an expereinced personal injury or insurance lawyer in your area. I would not rely on the advice of the claims adjuster who is handling your case. They will almost always lean towards paying an entity back, even if you don't have to.
If you do not have an attorney, you should get one as soon as possible to advise you on these complex lien issues. Your liability to pay off these liens could be found in State statute or in the contracts of insurance or benefits themselves. Retain an attorney if you do not have one.
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Mr. Lundeen is licensed to practice law in Florida and Vermont. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Lundeen strongly advises the questioner to confer with an attorney in your state in order to ensure proper advice is received.