If your spouses medical debts were incurred in Oklahoma you are jointly liable/responsible for them. Hope this is helpful.
You don't tell us who "they" are that is telling you to pay your Husband's medical bills, and the title of your question asks in which States you don't have to pay your Spouse's debts; so, I'm not completely sure what your specific facts might be.
However, most States, including Oklahoma, have what is called a "necessaries" statute. Necessaries are the things that are truly necessary for life -- food, clothing, shelter, basic transportation and medical care (are examples). In Oklahoma there is a statute that imposes liability on one spouse for the provision of necessaries to the other spouse. That can make you liable for paying your spouse's medical debt even if you didn't sign for it. There are limitations on that, though.
Also, you asked "which States" don't impose this kind of liability, so I wonder if the debt was incurred in Oklahoma? Would another State's statute be controlling?
Also, as I said before there are limitations on this liability which are very fact specific. From your question it sounds like you might be dealing with a debt collector on the phone who may or may not appreciate the limits of this kind of liability.
In other words, the answer to your question is that Oklahoma does require spouses to provide for each other's support, which can include medical care. That does not, however, answer the question whether or not you are liable for any particular bill.
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