I just bought Fallout 4 GOTY and after installing it and creating my ini files with the desktop icon I went to start the game with f4se. I got as far as new game but when I clicked on "New Game" what I go t was an infinite black screen with that little icon in the bottom right corner spinning.

I launched the game with the desktop icon, my mod manager, Fallout 4.exe, Fallout 4.exe as an Administrator all with the same result..

I know at least the problem with the infinite black screen (from Steam). For me it always helped to start the game in windowed mode instead of real full screen. Try that out (change the setting in the launcher).

More adventures of having the DRM edition on the same system as the GOG.

FO4 has never had native widescreen support, but good news, either "trueby9" or "widescreen interface" should fix that for you... I've used the former for years, but the latter looks to have better support and compatibility now.