Have a few suppressors, all in a trust.
Moving from AZ to SC.
Do I have to notify them before the move or just after?
Will I need to redo the trust?

You’re supposed to get the approvals prior to the physical move, but you wouldn’t be the first to submit the address change after a move. ATF Form 5320.20 is the form to complete - you can list three items at a time, and since your items are all on a trust complete the form(s) with the exact information shown on your approved Form 4s

If a permanent move (for the foreseeable future at least), just check ‘No’ in box 2.

For silencers, you don't need to notify at all.

For everything else, it's preferable to do it prior, but in reality I wouldn't worry too much as long as you've filed your 5320.20 at some point.

it doesn't matter which state your trust was formed in etc. you won't need to redo anything

quote:

For silencers, you don't need to notify at all

I would vote that is incorrect. The Form 4 approving your possession says "SHALL NOTIFY" in the case of a change of address.
And in any case I would personally want the NFA record to have my correct current address for lots of good reasons.

hrcjon is correct about the language regarding change of address notification being mandatory and not restricted to certain types of weapons.

For what it's worth, however, 5320.20 is NOT required for suppressors.

The Form 4 states:

If the firearm identified in item 4 is a machinegun, short-barreled rifle, short-barreled shotgun, or destructive device, the registrant may be required by 18 U.S.C. § 922(a)(4) to obtain permission from ATF prior to any transportation in interstate or foreign commerce. ATF Form 5320. 20 can be used to request this permission.

18 USC 922(a)(4) states:

(It is unlawful) for any person, other than a licensed importer, licensed manufacturer, licensed dealer, or licensed collector, to transport in interstate or foreign commerce any destructive device, machinegun (as defined in section 5845 of the Internal Revenue Code of 1986), short-barreled shotgun, or short-barreled rifle, except as specifically authorized by the Attorney General consistent with public safety and necessity;

While the obvious argument is "it's 'just safer' to do a Form 20", I would submit that it might be "just safer" to wear water wings everywhere, but if you're not going in the pool...

So to sum up, yes you need to notify, however unlike other NFA items you do not need to submit a Form 20 and get it approved before you move.