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You could find a card that has "virtual account numbers". I have a card that allows me to generate such card numbers. They're only valid for a single reseller, and they only last for two months, tops, so that may not solve your actual issue, but it would prevent you from handing out your real account number, and I can't see why they'd need to issue you a new card if one of the "virtual account numbers" was compromised.

They also generally let you specify a maximum amount for the virtual account, so when you create one to pay for a $50 charge, you can limit the charges to $50, and any subsequent attempts to make charges on that account will ultimately fail.