No. Given the facts you state, you would not have any personal liability for the business credit card debt. Even though California is a community property state, and generally each spouse is liable for the debts of the other, if the credit cards are in the name of the business, then the business would be primarily responsible, followed by your husband (or his estate). Even if the creditor tried to come after you for payment, as you have indicated your income would be exempt and protected. At best the creditor could file a claim against your husband's probate estate, if any, but if so you would just have to deny the claim. They would have no other recourse against you personally unless you were an authorized user and incurred the debt or otherwise signed any sort of personal guaranty on the account(s).
You are not responsible, but his probate estate may be. Even if his business is a corporation, it sounds like he is personally liable unless he took all appropriate steps to make sure only the corporation was liable.
Upon his death, the executor of his estate is supposed to take steps to determine his debts and to pay them if there is sufficient property in the estate to pay them.
If the debt is substantial, it might make sense to discuss bankruptcy with a local BK attorney.
If you are a co-signer then you are responsible otherwise no. If the debts are considered community debts you could be responsible for half of them. If the letters are comming in his name just mail the creditors a copy of the death certificate and tell them to stop sending letters. If you really wanted to be super sure you could file a Chapter 7 in your name and list all of his debts as a co-debtor but I do not think that is necessary.
The previous answers are correct but it's important to determine whether the credit card was actually for the business. I had a case where the credit card was a "business card" but the majority of the expenses were personal. This type of commingling can muddy the waters and create a possibility for attachment. If there is a trust involved, it would may be best to use Probate Code Section 19003 to serve notice on the creditors. If there is no trust involved, probating the estate will determine the ultimate disposition of the creditor.