What version of Electrum do you have?
When I installed my electrum wallet I am pretty sure I haven’t activated the 2FA mode, in fact I don’t have anything related with electrum on my authenticator.
You are probably referring to Google Authenticator. Electrum uses third-party service -
TrustedCoin for 2FA wallets.
Restore from seed? Should I try to install another electrum wallet from seed on other machine to preserve the one I already have on my machine to be safe? Should I save my private key before restoring?
There is no need to export anything or use a different computer. Try recovering your wallet using the seed. Use a different name for your new wallet file.