Im never use sms verification at Blockchain.info, if you afraid with your security, i prefered you to use coinbase.
Im use coinbase the sms verification is fast, also you can use calling to verification, but you can't import or export your wallet, it's 3rd party wallet.
I wouldn't recommend doing this , OP.
If you want to fix your issue then contact the support but don't switch to anything else (at least not web wallets) because the others like Coinbase don't give you access to your private keys which could get you screwed in case something wrong happens in Coinbase side .