What version of Electrum do you have?
Version 3.3.8
You are probably referring to Google Authenticator. Electrum uses a third-party service - TrustedCoin for 2FA wallets.
I am specifically prompted to insert my Google Authenticator code
Try recovering your wallet using your seed. Use a different name for your new wallet file. It is very likely going to generate the same wallet.
Do you mean I should create a new wallet within the application with seed original seed I have? Would this operation keep the former wallet I have on the app (to be safe)?
Thanks for your help!