getting rid of my "payment" options

The credit card associated with your iTunes account is the credit card you entered when creating your iTunes account and is required with an iTunes account. This isn't on your iPhone, but it is included with your iTunes account when purchasing a paid app from the App Store or when purchasing music from the iTunes store - which you have control over. If you want nothing but free apps, don't select and download a paid app. If an app is free or paid is not hidden, so don't select a paid app to be downloaded, which is entirely in your control.

Setting up a new account without an associated credit card might do what you want:
http://support.apple.com/kb/ht2534
But trying to sync apps from two different accounts might be a HUGE hassle.

I'm guessing Apple doesn't let you just delete your credit card info from your iTunes account, but you could try.