A spouse is generally not responsible for the debts of a deceased spouse unless the debts were incurred jointly during the marriage (so long as the married couple never resided in a community property state).
The estate is responsible for the deceased's debts. But since the spouse typically inherits the estate, as a practical matter, the surviving spouse's inheritance is diminished due to the debts of the deceased spouse.
However, there is a common law called "necessaries doctrine" that has held husbands in Florida liable for debts incurred to support their wife. As in all legal matters, you will have to consult an attorney in order to get a legal opinion that can be applied to your specifc facts and circumstances.
I hope this information is helpful.