If you aren't getting the 'none' option when creating an account then did you follow, exactly, the instructions on the HT2534 page when creating it ? I tried it yesterday on my phone in response to another thread and I got the 'none' option on my phone and computer's iTunes when creating new accounts on them
e.g. using a computer's iTunes : https://discussions.apple.com/thread/5729572?answerId=24321860022#24321860022
using iPhone : https://discussions.apple.com/thread/5847690?answerId=24700173022#24700173022
For your old account you should be able to change it to 'none'. On your computer's iTunes you should be able to edit your payment info by going into the Store > View Account menu option and logging into your account, and on your account's details page there should be a payment link. If you are doing it on your phone then try tapping on your id in Settings > iTunes & App Store and selecting 'View Apple ID' on the popup - that should also give you a payments link on your account's page. If you don't get the 'none' option on then see if anything on this page helps : https://discussions.apple.com/thread/4197488