i use aseprite! where the process is (1) generate a palette with palette dropdown > new palette from sprite (2) index the sprite to the palette using sprite > color mode > more options > floyd-steinberg error diffusion
libresprite probably has a similar flow. i couldn't say for any other art programs, although they probably have options for similar effects
there's also a really cool free program called img2pixel that does a similar thing
Omg your so welcome!!! This means so much to me! I'm so glad I helped too :)