I had this issue as well. I had been playing since it came out, with no issues. Tried to sign in on Friday, and it wouldn't log into the servers no matter what. After lots and LOTS of searching, this is what fixed it for me: (a post from another forum)
"Is Origin Installed on the same drive as DA:I? I had to reinstall Origin on the same drive as DA:I to get it to work.
Once I reinstalled origin to my other drive that my game was on, it fixed it.
I hope this helps with others and I'm sorry in advance if it doesn't help with yours.
*EDIT*
I just saw that you've tried this already. Sorry, for not reading all the way through what you'd tried already.
