I want to clarify that it's made clear on the back of our boxes that an account is needed. This has been standard for most PC games for a good few years now.
On to resolving your issue, the first thing to note is that you can't use the code a 2nd time, it was consumed on use with the initial account. This shouldn't be a problem though, you should be able to just log into Origin using the same account that you used previously.
When you're logged in and you try to play, how are you prevented from doing so? Do you get an error message?
Thank you.