It's common for us to request a copy of the entire trust, especially those sections which refer to the death of the Trustee and what follows. In some cases, it's typical for a revocable living trust to become irrevocable upon the death of the grantor/trustee, so you (or someone) will need to review the document very carefully. We have a corporate attorney who is available to review these if needed.
You should always require the proof from the individual requesting funds - if they can't provide it to your satisfaction, I'd refuse to do anything until they can provide it. That would also include, of course, the death certificate of the deceased Trustee, as well as the proof that they (or someone) has been named the Successor Trustee. You'll want to watch, too, for "Co-Successor Trustees" - that means there's more than one, and they have to act together, unless the document states otherwise.
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Opinions are mine and never to be taken as legal advice!