If the trustee files, publishes, and serves notice as set forth in Section 19003, then:
(a) All claims against the trust shall be filed in the manner and within the time provided in this part.
(b) A claim that is not filed as provided in this part is barred from collection from trust assets.
(c) The holder of a claim may not maintain an action on the claim against the trust unless the claim is first filed as provided in this part.