Account was hacked and my Origin ID was changed

EA Support should be able to help you with this. However, you should contact them from a different, 'clear' account. Matter of safety. Before you contact EA Support prepare some information:

A serial number from one of your games (if possible), it's the easiest identifier that can be used to locate an account. If you do not have a serial number available your Gamertag, PSN ID, or Character Names from online games will have to suffice.
The email address you would like to be contacted at
The original email address for your compromised account
Your actual date of birth
The date of birth used to set up your account

In order to contact EA Advisor you will have to follow these steps:

1 Click on the following link: https://help.ea.com/contact-us
2 Make sure that you're signed in
3 Enter the name of your game (in your case it will be Origin) and click Next
4 Select the platform you play on and click Next
5 Select the topic (for you it will be Account Security) and describe the issue
6 Select a contact option. Note that you may have to wait for a while.

Also, once it's all done and your security question/hack problem is solved, remember to change the password again.