UE4, Unity and Godot all have tools to get the PSX look down more or less accurately nowadays.
Sure, although note that I have yet to make anything with any of them myself so I can't say which option is the best. I've only read up on it since I intend to move from 2D to 3D dev soon-ish.
For shaders there is PSXEffects which seems very complete in the amount of things it can do. (There's two other paid ones in the asset store, but this one lists a lot more features.)
https://assetstore.unity.com/packages/vfx/shaders/psxeffects-132368
There is a free shader that has less things in it but might be good to test the waters of that style.
https://assetstore.unity.com/packages/vfx/shaders/psx-shader-kit-183591
Then there is an entire render pipeline created by a guy in the Haunted PSX community that might be as good as PSXEffects but might also be harder to set up (but is free)... but like I said, I've yet to use any of these, I've only made 2D games. Best to just try them out I guess.
https://github.com/pastasfuture/com.hauntedpsx.render-pipelines.psx