So I moved and had a few boxes of “important papers” ended up having a flood in the basement and I’m fairly certain it ended up getting tossed.
So I call the NFA and they say I need to email them my request and that they will send a copy in the mail.
So I follow their instructions to the letter.
I get a form in the mail to update my address. No copy of anything. I was under the impression that I didn’t need to update my address with them unless I moved across state lines?  Am I wrong?  
What should I do?

Since you lost the form and you moved...they probably need an updated address with new form you will get...I don't see why it would be a problem to fill that out and send it back.

Your old approved form will be voided with your old address...

In this type of situation, you should update your address as well as request a copy of your stamp.  You have a compound situation going on now, just get it all taken care of.

Fill out the form so they can send you the copy.

and when you get the replacement form, store multiple copies in multiple places. I also keep a scanned copy on my phone, and spare copies in the safe at my trustee's house.

You will have to send a 5320 form in to get a certified copy at a address that is not on the form4. I had to do this when they lost my original in the mail and I moved the same time