Legally you're only required to send the form for the SBR, but I would do the 5320.20 for all the NFA items since it's a permanent move. It keeps their record updated and it helps prevent headaches if you even plan to sell/transfer the items in the future, which as you said you plan to do apparently moving them to your other trust.
You'll have to do a separate form for each trust/entity as well.
As for updating anything on the trust document itself, that can depend trust to trust. If the trust lists some address as an identifier for you or others listed in the trust, you might want to type up an amendment clarifying that you moved to the new address. Also check to make sure that the language of the trust is valid in your new state, some states have unique laws about trusts in general.