The form to transport across state lines is a 5320.20, whether for transportation or for permanently moving. For moving you check "no" to the question "firearms to be returned to original location".
Most of the ATF NFA Title II forms are "5320s". A "Form 1" is actually a 5320.1, a "Form 4" is a 5320.4,etc.
Note that a 5320.20 is NOT required for transporting suppressor. But a lot of people do it anyway, and ATF still returns the other copy and processes it. Personally, I'd definitely do it for a permanent move.
To me, the big concern over moving with a trust suppressor is whether the trust is still valid in the new state. If it's registered to you, personally, no harm no foul. But if a trust is made invalid by different laws in the new state...that I'm not sure about. For example, if you moved from a state that allows "self dealing" (ie using the suppressor as the trustee before giving it to the benefactor), to one that does not, I'm not sure if the invalidation of the usage under the trust would be an issue or not. That's an attorney question.