Actually, they should have cut of access to your online account too. But having a look at their website, it seems that it is poorly developed. Consider this,
In case you have upgraded your SBI Card, please register again with the New Card details,
this implies they are maintaining records on the basis of Card Number rather than Customer ID and link all the card accounts.
For your safety,
1. Preserve a printed copy of the e-mail as well as the e-mail itself.
2. Log in to your account and print a statement which will say ZERO Balance due or likewise
3. Finally send a Registered AD Letter to the bank that my A/C has been closed with all dues PAID OFF and you are not liable for any transaction on the Card A/C Number. Save the Letter and Acknowledgment and you are all set.