I'm the same way I like paypal to mask this. However some credit cards now have "virtual cards" that give you a temporary online number that you can use for specific transactions. It's another great way to avoid your data getting out there. I use one from Citibank as I have a CC from them.

I highly recommend that instead of Paypal too if you suspect a site will have a much higher chance you'll need to reverse a charge. Paypal has longer wait times for disputes than the CC companies. CC companies will also have your back a bit more, typically.