I recommend not bothering. Liability for fraudulent CC activity (in the US) is very low (max $500), and banks never enforce the liability for two reasons: 1) it's easier for them to chargeback the transaction to the vendor, thus they're not out of pocket, and 2) it causes customers to leave for a provider that doesn't hold them liable.
How much of your time spent chasing down and changing a CC number is worth a maximal risk of $500 with a vero low probability of occurring?
I'm pretty sure you're off by a factor of ten. If someone makes fraudulent charges with your credit card, your maximum liability is $50, and you'll only have to pay that if the charges occur after you've reported the card stolen. (There are different rules for debit cards, that doesn't apply here.)
How much of your time spent chasing down and changing a CC number is worth a maximal risk of $500 with a vero low probability of occurring?
EDIT: Thanks to nenolod for the correction!