I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your situation.
The company that you're negotiating with unfortunately does not have to accept payments in installments. However, they will sometimes accept SIGNIFICANTLY less than the total owed if you can come up with cash upfront. That might be worth looking into, if you can scrape any money together (because they don't really want to have to sue you).
As a practical matter, if they get a judgment, they won't be able to collect it all at once unless you have the money lying around. They'll be stuck garnishing your wages and getting monthly or biweekly payments. They'd get 25% of your post-tax earnings. So you can trying pointing that out to them and seeing if they're willing to accept payments now in exchange for saving the time and expense of going to court, considering that you do not have the cash and they won't get full payment even if they sue you.
You also have the option of suing the other signer for any monies that you wind up having to pay because they stopped paying, if the agreement was that you would cosign and not have to pay anything. But you may also have to settle for collecting your money slowly over time.
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