Do you owe my deceased spouSE debt?

Depends on the State, however any deceased persons assess would be drawn down on their assets through probate. This means that if you held joint assets with the spouse; ie., home, banking, investment accounts, then what every value therein is up on the table for the creditors. In those States that have community property laws your at 100% risk of loss, however non-community states your risk is 50%.