Take a look at these two examples from last year's jam. I think they both did the sound very well.
https://clonedeath.itch.io/sokoban-nokia-3310
https://machinescreen.itch.io/hwy3310
If I am understanding your P.S. correctly they do not have to "snap". You can see the pixels move relatively smoothly in this gameplay video Jannick provided for last year's jam https://www.youtube.com/watch?v=32PsXnU1070&feature=emb_title
I hope that helps let me know if you need any more clarification!