New copy of Mass effect 3 will not register its product code on origin

Sorry to hear about this.  Unfortunately there's nothing anyone here can do for you.
The AHQ is for "player-to-player" technical self-help and game-play discussions.
We're all just fellow gamers, doin' the best we can...

You need "official" support from the EA Game Advisors.
They are the only ones who can resolve account, code, or game/DLC authorization issues.

Contact them using the Help Center.

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After getting to the Help Center and filling out the required information, (Product/Problem/Platform)
look for the "Chat Now", "Call Me", or "Email Now" buttons, not the AHQ "Ask Now" one.  (You've already done that.)

NOTE: The options you see may vary depending on your region and the time-of-day.

If you select "Chat Now", be sure you have enough time (at least one hour) to wait before being connected to someone.  😕mileyfrustrated:

Good Luck!