If I update my offline cold wallet will that erase my current wallet data? It's backed up but I don't want to be diving into my backups if not completely necessary (lots of security).
No, it'll still use your existing wallet. By default you can see your wallet files in "%appdata%\Armory" on Windows. Create a copy of that folder on the offline machine if you're really worried?
Do I need to create a new wallet file (and new addresses) within Armory 0.96.3 in order to enable segwit or will my old wallet and addresses be adequate after the version update? How is this process supposed to work?
Nope, existing wallet - just update your preferences for receiving addresses and (potentially) your change address to use P2SH-
P2WPKH (segwit - they'll start with a 3 rather than a 1).
Sending right now just to use segwit won't really save you on fees overall... so when fees drop to a level you are happy with send your existing coins over to segwit addresses in your wallet. For new coin purchases/transactions, you should send to a segwit address rather than the old format wherever possible.
As for the software upgrades, Bitcoin core on your online machine needs to have the segwit/witness flag - The latest does, but has been supported for a few versions now.
I'm going to quote a reply I made recently on this which is mostly relevant for you, but you really want to make sure you upgrade to
> 0.96.3.99/0.96.4 RC1 from the pinned thread <. See below:
Unless you are worried it is compromised, you can continue to use your current wallet for segwit.
Ideally, you'll want Armory 0.96.3.99/0.96.4 RC1 and then change your preferences to use P2SH-P2WPKH (Segwit) in Armory. Change address too, perhaps but that is up to you regarding privacy.
I don't think upgrading Core is required, but the node rpc does need to be running on your full node. Offline machines with 0.96.4 RC1 can force segwit availability with --force-enable-segwit.