There's quite a few retro shooters like that nowadays.
Prodeus, Ion Fury, Amid Evil, Project Warlock, Project Downfall, Hedon, Viscerafest... and those are just the ones I know about. If you include retro 3D (think mid-90s style) there's even more!
There's also a lot of mods and TCs for some classic games, like Doom or DN3D.
You should follow some gaming channels on YouTube that focus on retro shooters, they're a good source of new ones to play. I'd recommend ICARUSLIV3S and GmanLives if you'd like to find more.
I love this renaissance the genre is having, combining older aesthetic with better art design and lighting. Valheim comes to mind. I hope janky looking games but that are as deep as Deus Ex aren't far behind.
>And what I love about them is that they're able to just use the original engines* from decades ago
DUSK is a great exception to that, where they actually used a very heavily modified version of Unity to get something similar to ioquake or the early DOOM engines. IIRC most of the graphics programming was actually done to stop Unity from making the game "look better" :)
Amid Evil is another notable exception that I can think of of the top of my head: it uses Unreal4.
Oh man, Amid Evil is pretty crazy in that regard. All the game elements are sprites with depth mapping, so they react somewhat realistically to level illumination.
Prodeus does something similar as well. In fact, you can switch between 2D sprites and 3D models on the fly.
Other than games others recommend, maybe check out the mod community of those games you love. There are TONS of mods, and I mean top quality mods of classic FPS, enough to spend a whole year or more to play all of those through.
I saved a few Youtube channel for these kinds of things:
I actually designed and built a (modern?) game like this!
If you have an iPhone you can check it out. It's sort of "Portal meets Doom" https://blog.sb1.io/gateescape/
I used some screenspace shaders for the controls and color blending, but the engine itself is an oldschool raycaster. Building it posed a lot of interesting problems like how to recessively render sprites through portals and things like that.
For something a little different, check out Sonic Robo Blast 2. It's a third person 3D Sonic the Hedgehog fan game based on the Doom Legacy engine, and it's surprisingly quite decent. Certainly better than some of Sonic Team's efforts.
There are some out there. I was working on a game using the original Rise of the Triad engine which did 2D sprites, but the idea evolved to larger than that engine could really support. I'm now making a modern engine but keeping some of the philosophy to target low spec machines, so some of my "models" are 2D sprites.
Get FreeDoom and PrBoomPlus for PC, it's a vanilla Doom/Doom2 compatible Iwad which will allow you to play any Doom2 PWAD out there (and get an amazing shooter for free by itself, too).
I wish people would still make them instead of going full 3D!
EDIT: Thanks for the game recommendations everyone!