You can log into Origin and grab DLC for Dragon Age: Origins, Dragon Age 2, Mass Effect 2, and Mass Effect 3 by selecting the games in your library, clicking on Extra Content, then selecting Get it Free for each one.
You can also do it through your browser - Origin Game Library
Thanks for the heads up @carvalho20ptc
Considering these are versions EA and Bioware are no longer focused on supporting, this might be the best way to go about it. Sounded like a crappy system and at least those of us too cheap to get the DLC and remastered versions to get the originals for free.
Took me a while to find the browser method for game library, so I thought I’d include that link: