I had the exact same problem. I did two things which solved it:
1. In Origin right click on Mass Effect 3 and select: CHECK FOR UPDATE and then REPAIR INSTALL.
I can't confirm whether this part of the process is essential or not, but I did do it. But part 2 is key:
2. Log into http://social.bioware.com using your origin ID. Just doing this should help. But just in case look around for anything that synchronises your game and galactic readiness data.
Anytime I get the "CAN'T AUTHORIZE DLC" message I got to http://social.bioware.com and just log in again and it solves the problem. I've had to leave the tab open. It's tedious, but so far it's worked every time.
I did not need to reinstall Mass Effect 3.
I hope this helps!
