No requirement to notify if the move is within he same state. ATF does ask that y inform them so that they may keep the registry more accurate.
I notify so that I have a confirmation that they  received notification.

For no change of state, nothing is required.  However, ATF likes it when you inform them of the move.  If you want to do so, send a letter stating that the firearm (include make, model, caliber, s/n, and type- same info on your Form whatever) is being moved to a new address due to a household move.

Send the letter to:

Bureau of Alcohol, Tobacco, Firearms and Explosives
National Firearms Act Branch
244 Needy Road
Martinsburg, West Virginia 25405 USA

You could also notify them using a Form 5320.20 instead of a letter, though I guess it's technically an improper use of the form.  (A 5320.20 is for interstate transport of a NFA weapon, though I doubt they'd quibble if you use it for intrastate notification for a permanent change of address.)  Make sure to send the 5320.20 in duplicate if you do send one.

If you want them to notify you of receipt, request so in the letter.  That and/or send the letter certified, return receipt requested.  If you do a 5320.20, you must send it in duplicate (as with all other forms) and they'll send one back to your original (old) address.

FWIW, you don't ever have to notify BATFE of any intra move at all.  Same county, different county, makes no difference.  They only care about the state when it comes to moves.  You only need to notify them of a permanent address change across state lines on a 5320.20.  And you don't even need to do that for a silencer or an AOW.

FYI,

Mike

Eh, do the 5320.20 in duplicate and send it in.  Might as well put both of them on there and just be done with it.  Nothing like having an official stamped document from ATF acknowledging the move in case of stupid down the way.

Mike

You don't even need to do a 5320.20 to take them out of state, unless it's a permanent move.  Save a tree and reduce the paperwork.

This is a bad idea.  Not only are you misusing the 5320, you are misusing NFA Branch resources.

The 5320 is not a change of address form, it's an application for interstate travel.  You imply the change of address by indicating on the form that the nature of the interstate move is permanent.

They don't provide an official form because it's not a requirement to notify them of intrastate changes of address. Submitting a 5320 application mistakenly identifies your request as a higher priority action when such requests would more properly come in the form of correspondence.

If you feel you have to tell them that you moved down the street, then send a letter stating that.  Don't present it as something else.

We're all waiting 5 months now for approvals.  Wasting their time is wasting our time...

Correct. I choose to notify but it doesn't really matter if you do or not.

To be clear,

I can take my suppressor from FL to GA without notifying anyone, correct?

Yes but I recomend that you do a 5320.20 before doing so.
Some police officers might not know that its legal to go over state lines with suppressors & AOWs without an approved 5320.20.

Remember as far as the law goes, a suppressor is a gun