Since you downloaded the blockchain already: you can create a transaction from Bitcoin Core, and copy the raw transaction from your offline PC to your online PC for broadcasting. Enable Coin Control in Bitcoin Core, and make sure no change is sent to a new address (if you do, that address may or may not be included in your old backup).

Using Electrum should work too (see my earlier post), but in that case you didn't have to download the full blockchain. Keep track of change amounts: if you don't spend the full amount, make sure you send it back to your own address.

You'll also own Fork coins, but don't do anything with them as long as there's Bitcoin on your addresses.