I've recently bought a new PC and I am facing the same issue I had on my laptop. I remember that I have actually found a solution for this before, but I cannot find it anymore. So basically, I can't use Push To Talk while running a game as administrator, so I have to use Discord as admin too. Well, running Discord as admin doesn't let me drag and drop files and I always have to either choose them manually, or just restart whole app.
As I remember, there was a fix that could be done by using Regedit. (I used it on my Windows 8.1 laptop, and it worked)