Why isn't my iPhone using my new Apple ID?

I just created a new Apple ID but my phone isn't using it even when it says it is in settings, and at the bottom of the Apple and iTune's pages. I tried updating my apps but it asked for my password from my old ID? I need to creat a new Apple ID because in my old one it started asking for credit card information which I didn't want to put in anymore and without that I couldn't buy free apps or update them and sadly the none option is no longer available.

I think you're having the same issue that I did when changing Apple IDs. New downloads will use your new ID but apps that you downloaded previously will always use the old one. The only solution that I know of is to delete the apps and then download them again so they become associated with your new ID. If all/most of them are free, its not a big deal and actually a good way to clear up things that you no longer use.