I have no problems with Fallout 4 and drivers 16.9x-16.10x.I think your problem is not driver version related.
I had similar issue, when i tried install amd chipset drivers before vga driver. It is bug of AMD Catalist Install Manager.
You can try install video driver manually via device manager. But first, try to restore your system to restore point to a date, when your vga driver was present.
Or use DDU and try latest driver again.
Show us picture of your device manager (video adapter).
Well since my last post I came to the conclusion that I had to wipe everything and re install fresh windows 7 from my disk, so I did and I rolled back to my old 7770 gpu while I was at it, it was much more stable - I ordered a new 1050 ti (nvidia, here I come!) because I am totally put off amd after all this, I want my money back but I know I wont get it... anyway after a completely fresh windows install I was finally able to actually put drivers on (ooooo) and can now play a few old games while I wait for the good gpu to arrive, count me as never coming back amd, I hate being this salty but that's what you did to me. Very unhappy