There are two ways to get overpaid in unemployment, there are fraud overpayments, where you actually lied or falsified your claim in some way to obtain benefits. As in "No, I did not work any last week," when you did. Or "I was fired unjustly" or "I was laid off" when you really quit. Or, 'I'm healthy and I've made these work searches" when you aren't and haven't. Or you forgot to mention that although you were laid off from your day job, you worked at a club in the evenings and made another whole set of income and filed for unemployment while doing it.
Fraud overpayments are usually caught by wage records cross matching, if you work for another covered employer, or through the information submitted by the employer, or by income tax records of earnings or by someone calling in to report you for committing unemployment fraud. When they are found, they are investigated. The claimant is contacted, the investigator researches and then makes a decision whether the fraud actually occurred, how much fraud occurred, and usually penalties are attached to the overpayment.
Then it goes to the legal arm of the agency for collections. They want their money back. Far more than they want to prosecute, they want that money back. They generally will not go away, whether you have anything for them to get or not. This overpayment that you owe to this state agency will be around for a long long time to haunt you.
Or there is what is called "non-fraud" overpayments, where you file for benefits, your initial decision is made, approving the benefits, and you begin receiving them. Then the employer appeals, the second in person hearing is held, and the decision to give you benefits is overturned. Those benefits you have received are an overpayment now, you are expected to pay them back. But they were not obtained fraudulently. They may go on your record, but they are not generally pursued aggressively. There are no penalties involved. They may eventually begin collection processes on these kinds of overpayments, such as taking your state or federal income tax refunds, etc, and you would not be able to draw any more unemployment benefits until this was repaid, but they seldom take non-fraud overpayments to collection aggressively.
In neither of these two scenarios did the agency "make a mistake." And I also am having a seriously hard time seeing how it got to this point unless it was fraud and you had ignored their initial inquiries. Even though they cannot "get" your husband's property, perhaps, they can certainly keep coming after your and his tax returns, any income you make, etc.
If the "past due" amount is $6000, and you are not going to be allowed to negotiate or set up a repayment program with them, then I gather you must have done so in the past and failed to make the repayments? In the big scheme of things, $6000 (which will probably grow as time goes on and interest and penalties accumulate) which you did get at one time and now can't pay back is a relatively small amount to drag with you through life and have affecting your credit and your situation for years to come. Get a job and pay this off.