You did not give us many facts, but in general ...
NO, you do not have to pay off your spouses debts upon their death if they are not also your debts too (i.e. if they are NOT joint debts).
If he incurs debts on his own (or did before marriage) ... say for boat, car, truck and house ... and you marry him, you do not owe for those debts normally. However, if you co-singed with him on any debt (before OR after marriage), then you become liable with him for those debts.
Ordinarily, upon a persons death, their debts only get paid off of of their assets at death ("estate") ... and ordinarly nobody else has to pay off their debts if their "estate" is not worth enough to pay off their debts. However, that does not mean that you would not have to pay off the remaining debt on a JOINTLY OWNED asset where you signed as co-borrower with the spouse (his death would not wipe out your liability, it only wipes out the decedant's liability on the joint assets).
There could be some unique exceptions to the above, so if you are really concerned about this, you should see an attorney who does wills and estate planning who could not only advise you, but might be able to help you arrange debts and titles to properties in a way to minimize any exposure you are concerned about.
ROB
rob