This is probably too late to help anyone, but in case someone has Photoshop but dismissed it:
I've been using Photoshop to export the final versions of all my Playdate graphics, because it's got the Mac Classic-style pattern dithering that I haven't seen elsewhere. I don't like the default handling, but going to File -> Export -> Save for Web (Legacy) and using the settings shown here has been giving me exactly what I want.
Note that because my game is cartoony, I'm mostly using 5 shades of black, white, and gray specifically chosen to give the familiar dither patterns. For an arbitrary image with gradients like the UFO, pattern dithering gives this:
while Diffusion dithering gives this (which I'd assume is more what you were looking for, and is similar to the one that eda posted):
I've never seen options in apps like Acorn or Pixelmator or Affinity Photo to have as much control over the dithering.