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What happens when you update apps ? Are you getting any errors ?
What exactly happens when you download apps ? Are you getting any error messages ?
What exact error are you getting ?
Please tell me the error or message you are getting when you updates apps, that will help me to identify what the problem is.
There are different errors and each have different solution
Each application you downloaded using a particular Apple ID is tied to that ID/email address. That means whenever you try to update those apps from App Store, it will still ask for the old Apple ID/email regardless of what ID you signed on your iPhone now. So what you want to do is, just delete those application and redownload them using your new ID/email and then that application will be tied to that account.
Deleting the app will also delete it's contents. It won't delete anything of server based apps like Facebook, Emails etc.
To delete, just tap and hold down the app and it will start wiggling. Tap the x icon on the app and it will be deleted. Then open App Store, search the application and download it.
Important : Go to Settings select iTunes & App Store and make sure you are using your new or active ID there. If you see old Apple ID there, then tap it and tap Sign Out. Then sign in to your new or working Apple ID.
Let me know how it goes.