I hadn't played Dragon Age Inquisition since early November and I tried to start it again a few days ago and got the same message you did. I've tried a variety of things to get it to work and haven't had any success. You may have read a variety of tips too, but I was able to get to the main menu of the game by pressing alt+enter (I frantically pressed it repeatedly but it was probably unnecessary) immediately after launching the game. It only worked if I was able to do it before the DA:I window popped up. However, once it reached the game menu the second I clicked on it, it crashed and showed the error message again. I've sent in feedback to Microsoft because I tried rolling back my drivers and that doesn't seem to be the issue.
Also, did it begin happening after a Windows update? I'm on Windows 10 and I was looking at past update info to see if there was anything there about DirectX or something. On the 11/14/17 update for Windows 10 (OS Build 16299.64) it listed "Addressed issue that causes a black screen to appear when you switch between windowed and full-screen modes when playing some Microsoft DirectX games" as an improvement/fix. I'm not tech savvy in the slightest so this could be completely unrelated but I thought that was interesting since I can get DA: I to the menu as long as it's windowed but when I click on it, which usually makes it go fullscreen, it crashes with the DXGI error.
https://support.microsoft.com/en-us/help/4048955 this is where I found the info on the Windows 10 update.
Also, I've never posted anything on here before so I apologize if something's screwed up.