This wouldn't be an issue if credit card providers offered a standardized system for making subscription charges. When you sign up, you'd pre-authorize them to charge your card a certain amount per time period. You could revoke the authorization through your credit card company's website, along with your other subscriptions. If the service wanted to change the price, they'd need a you to re-authorize it.
If they were doing that, they should also overhaul the whole system so you're never giving your credit card number directly to the sites. Similar to how if I use OAuth to log into a site, that site never has a chance to see my password.
