Am I responsible for my mother's debts now that she's passed away?
Just because you signed your mother's checks for her, using her money, does not make you personally liable for her bills or debts, unless you signed a document specifically agreeing to be personally responsible. However, the fact that your name appeared on some of the checks might make the care center mistakenly believe that you took personal responsibility for your mother's bills. You should immediately write to the care center and tell them that you were merely handling your mother's finances for her, but that the money was coming from your mother's account and you never assumed personal responsibility for her debts.
Be aware, though, that the care center might still look for payment from you as your mother's heir. If the funds which your mother still had when she died -- known as her "estate" -- passed to you, the care center has a right to ask that your mother's debt to them be paid out of those funds. If the care center asks for payments out of your mother's estate but you received no such funds from your mother's estate, you are not personally responsible for her debts. If that's the case, notify the care center that no funds passed to you from your mother's estate and that you are therefore not responsible for your mother's bills.