In general debts of a deceased person must be paid out of his or her probate assets. So your husband's debts would be paid out of any assets that are in his name alone.
You should consult with an estate planning attorney to make sure your assets are properly owned. Pay particular concern to any real estate owned and have an experienced attorney review the deed to make sure it is joint with right of survivorship and not just owned jointly. Joint with right of survivorship will not be a probate asset. If it is held jointly then creditors could force the sale of your home to pay the debt.
As far as the liability goes, you are not liable for debts of your spouse unless you guaranteed the obligation, or signed as an obligor on the credit card contract. Without being glib about it, it is relatively rare that a credit card lender can come up with the contract, which often never existed in the first place.
In Ohio a surviving spouse has first priority in the assets of the deceased spouse up to a certain limit. And often, estate assets are minimal or non existent by virtue of survivorship and payable on death accounts. You should sit down with a bankruptcy lawyer and look at the options to deal with this "huge credit card debt" now. Failing that, you certainly should spend some time with a lawyer who understands this sort of estate planning. But it certainly doesn't make sense to sit around and worry. The answers, and perhaps the solutions, are not that challenging.