Ohhhh. My bad. I misudnderstood what you wanted. That's what happens when you're trying to do two things at once. Sorry.
If I understand right, what you did was chose to save the data locally when you signed out of your old account. That caused a local copy of all yoru info. When you created and/or signed in with your new account, you should have had an option to Merge the data. If you said no, then each app would have created two separate accounts. The old one would be a purley local account that syncs with nothing. The second would have been a (probably) empty account that syncs with your new iCloud account.
Probably the best way to fix this would be to go to Settings > Mail, Contacts, Calendars, choosing that account and turn off the offending apps. When they ask if you want to save locally, say NO. When you turn them back on you should get a Merge option. Say YES to that. Whatever is on the new account (stored in iCloud) should merge with whatever is still in the local account, and both will sync with the new iCloud account. Keeping all you stuff in one place.
That's assuming you want to keep all your stuff. If not, the solution is likely to be more complicated.
