Can I sweep keys into an Electrum wallet? I'm still a few years behind.
Yes, that should work.
Technically, you could also just sign the tx and broadcast it. To that avail, turn off Core and make sure autobitciond is turned off in Armory settings. Then start ArmoryQt and let build its db off of the blockchain data
up to the most recent block you have on disk. You will get to see your balance for that height (some time in 2021/22 I presume). If your coins have not moved in the present, then they're still "there" in 2021 or whatever snapshot of the past you're working with. You can create and sign a valid transaction from that snapshot.
Use the offline transaction UI and once you have signed the tx, extract it in binary instead of trying to broadcast it from ArmoryQt. At that point, you can copy the tx binary and broadcast it via any web tool/explorer. If the tx is valid, it will mine.
This should make sense if you consider the blockchain is an append only ledger. If your coins have not moved since 2014, all you need is that data from 2014. None of the data added to the chain in the interim affects your coins, so you don't need it to create the tx.
I may want to keep Armory for myself Is there a version of core I should try to update to?
Any modern version works. You have to create/amend your bitcoin.conf to have the following:
Since you started to download the chain with an old version of Core, you should be fine to update Core so long as you amend bitcoin.conf first. You should not have to redownload any of the chain data.