Any chance you have the DX9 files running as administrator and not the DX11? In the FFXIV folder there will be two client .exe's for them. Also, do you have the Windows 10 power settings set to "High Performance?" It may be a longshot, but it sounds like your network card or hard drive might be putting themselves to sleep because they're not detecting activity. There's also a setting under your network driver to stop it from going to sleep to conserve power (which on a desktop isn't necessary). Googling that might help.

If that doesn't work then I'm stumped. I know a couple of newer AMD cards had hanging issues under DX11, but to my knowledge the 7850 wasn't part of that. It was mostly around the 285/380. Even still, running the newest drivers would fix that issue because they've had it all patched up since 16.3.2.