I had one running at 2560x1440 reliably, but that installation I must have deleted awhile ago.

So, I'm using the widescreen patch and currently running at 1920x1080, which looks very nice. But I want to get back to at least 1440P, or 2160P, preferably. So far, the game always crashes quickly (the old Page Fault) when I run at those resolutions.

You have to set a low resolution in-game, like 1280x720, and set a higher resolution in dgVoodoo2 or nGlide options. If you're not using either of those, you have to to get high resolution working reliably.

Late to the party, but if you get Undying Rebirth from ModDB, it can easily do 4K and uses Vulkan for rendering. Works like a charm for me.

Fair warning, if you used the Widescreen patch, you'll probably need to do a clean install; I tried switching over to Rebirth from that patch and it caused all sorts of issues until I deleted everything and reinstalled.

Think you mean Renewal, not Rebirth at least that's what works very well for me.