My computer is currently half-broken and I can only use it if I use the default graphics drivers instead of the actual Nvidia ones. I was able to play and complete Fallout 1 just fine despite this, but Fallout 2 has graphical glitches in the intro videos and then the main menu doesn't show up at all. I already tried using the Restoration Project, but that just left me unable to run the game at all. Is there any way I can get this functioning properly (apart from buying a new computer) or am I boned?

Fallout 2 requires 3D graphic driver with DirectX 9 support, so you're "boned".

The page for FO1 makes that claim as well, but I was able to run it just fine (using Fixt, maybe that was why?)

Precisely. It includes tons of content, including SFall's DDraw (Direct Draw), which is the part of DirectX which is concerned with 'drawing' things on the screen.