I remember learning this tune many years ago during one of my first jazz guitar lessons. I guess it's one of those Burrell tunes that are so cool that every guitarist wants to learn them. Just like "Midnight Blue" of which I posted the music and a backing track earlier here.
Anyway, when I returned to "Just a Sittin" last week I listened to the original some more and changed some chords and fingerings. I think what I am playing in the video below is closer to the original than what I had under my fingers for years. Hopefully you can figure it out yourself. Such a nice melody to play solo too.