I'm more then sure that you didn't lose your coins and you can get them back .
If you still can't send your bitcoins anywhere then I'd suggest you to restore your Electrum using your seed and that should cancel the Two Factor-Authentication part and you should be able to send your coins wherever you want to without any issues however if you don't have the seed somewhere then I'm afraid there isn't much you could do .
The seed won't be able to help at all.
To send transactions with the 2fa, you need to pay TrustedCoin, which is the service that does the two factor authentication. It should be something that is automatic when you send the transaction, and they will just ask you for the authenticator code.