I would handle the old offline Bitcoin Core as if it's paper wallets: get the addresses, and get the private keys. As long as it's offline, there's no risk.
Then, on an online computer, install Electrum, and import the addresses (or one at a time, for privacy). You'll get a watch-only wallet, create a transaction to your Ledger address. Set the fee at 1.3 sat/byte.
Copy the unsigned transaction to a USB stick. Note that you're still safe, no private key has touched the internet!
Reboot your computer, or get another one. Download a Knoppix Live Linux DVD. Boot your computer, DO NOT go online. Unplug cables, don't connect to Wifi. Start Electrum (it should be available by default), create a new wallet, enter the private key to the address you created a transaction from. Then copy the unsigned transaction into Electrum, VERIFY the transaction details, check all characters of your Ledger address (get the actual Ledger and verify it on it's tiny screen). If all is good, sign the transaction, export to USB stick, and turn off that computer. All traces will be wiped from memory.
Use your online Electrum to broadcast the now signed transaction on your USB stick.
Rinse and repeat for your other addresses.
I typed something similar earlier this year:
I was considering Electrum, only downside is that when creating a wallet, seed creation is exposed
Online:Install Electrum on your PC.
Import your address to create a watch-only wallet.
Preview the transaction, Copy the unsigned transaction. Put it on a USB stick.
Offline and running without hard drive storage:Get a Linux LIVE DVD. Use Knoppix for instance, or any other distribution that comes with Electrum pre-installed.
Unplug your internet cable. Close the curtains. Reboot your computer and start up from that DVD. Don't enter any wireless connection password. Keep it offline.
Start Electrum. Import your private key.
Copy your unsigned transaction from the USB stick, load it into Electrum.
CHECK the transaction in Electrum. Check the fees, check the amount, check all destination addresses (
character by character).
If all is okay, sign the transaction. Copy it back to your USB stick.
Turn off the computer. That wipes the Live LINUX from memory and all traces are gone.
Online:Use your normal online Electrum to (check again and) broadcast the transaction.
Bonus:After moving all your Bitcoin, and once the transaction confirmed, check if you own
Forkcoins.
Note: when in doubt, don't do it!