Are her children required to pay her outstanding medical and nursing home bills or can we just divide the estate and move on?

A decedent's heirs have no rights to the assets of the estate until after all the decedent's outstanding bills are paid. The executor of your mother's estate must gather all of her assets (real and personal property, bank accounts, vehicles, etc.) and notify all of the creditors (medical, nursing home, funeral home, mortgage company, utility companies, credit card companies, etc.) to submit a final claim by a certain date. All of those outstanding claims must be paid off first. The heirs take whatever is left over. If there is nothing left over, the heirs get nothing.