Read this article.
Bsically, yes you should report his death – may need to do it by mail and send a certified copy of the death certificate — by registered mail to them.
Make sure his name is not on any of your accounts or listings in your credit report.
If a 1099 comes in Jan, requesting taxes on unemployment benefits he did not get, don’t worry. he is dead. They can’t even try to collect, unless your checking, savings accounts or home are still in his name!! that should have been changed when you created your own new estate plan after his death.
Get on top of all that and you don’t have to worry!