You ARE NOT responsible for paying his bills as long as you were not on the account. This does not mean that they won't try to convince you otherwise - the best line is that you benefited from his charges (groceries, gas, etc.), therefore you're liable. Don't fall for it.
Whatever you do, don't agree that the debt is yours and don't agree that you'll take responsibility for it. If you do this, they could transfer the debt from your husband to you.
The creditor has a right to seek payment the debt, but the money is owed from the estate - not you. Thus, if/when you probate his estate, the creditor will have to file a claim with the court. If the estate has enough money in it to pay all of the bills, they'll all get paid. If not, they'll be paid on a pro rata basis.
The only way that the creditor can access your bank account is if it gets a judgment against yout husband, which it can no longer do because of his passing. Therefore, don't let them bluff you.
You've got rights against these creeps under the Fair Debt Colletion Act policed by the Federal Trade Commission. Here is the website:
http://www.ftc.gov/bcp/edu/pubs/consumer/credit/cre18.shtm
You can contact the FTC to report a creditor, file a complaint and understand your rights.
My suggestion is to maintain that it is not your debt, and you don't owe it.
Lawyer (JD)
Litigation Attorney practicing in multiple State and Federal Courts, Arbitrations and Mediations