If a Will says that the remainder of the estate be added to a Living Trust, do bank accounts in the decedent's name automatically go to the Trust?

I am so sorry for your loss,  It really depends on how the Last Will and Testament is written to determine how the account is to be dealt with.   Is the Will what is known as a "pour over Will" meaning that any asset not already in the trust name will "pour over" from the estate in to the Trust on the death  of the decedent? THat covers assets that the decedent forgot to change over to the name of the trust.  If not, then yes, you may have to probate the account (if small estate or regular that depends on the particulars here) and then once appointed as fiduciary transfer it to the trust. Please let an attorney in the area review the documentation.   Good luck.