Your question concerns paying estate bills before an inheritance is distributed. Specifically, you state that you mom has a large account" with my sister's name and in trust (itf)..."
I'm unclear what type of account your Mom has. I don't know what "itf" means. Is the account in a bank trust or in a pay-on-death form? I gather that the account is not in joint tenancy with your sister, because you state that the account is in trust.
However the account is held, your mother's estate is responsible for paying her bills after her death. A person has the right to specify, in a will or living trust, how his or her bills are to be paid"”what resources are to be used for payment. It seems unlikely that your mother has so specified any specific assets to pay her bills. Therefore all of her property would be responsible for payment. If the (trust) bank account is your mother's major, or even only, significant asset, the bills would have to be paid from that account. But if your mother has two or more substantial accounts, or other significant assets, it isn't clear which resource wold be used to pay her debts.
I suggest that this is a matter you, your mother and your sister clear up while your mother remains alive, if you possibly can.