Short answer, No, Pokemon that spawn running don't seem to have different behaviors when shiny.
While some shiny Pokemon do have certain behaviors suppressed, from what I've seen it's mostly for Pokemon whose species can despawn themselves in the over world, like the Voltorb line self-destructing, light Pokemon flying away in the wind, Sandygast melting in the rain, Pokemon like Dunsparce and Silicobra burrowing away (in fact the only instance I can think of for a mechanic where a Pokemon doesn't despawn is Bramblin whose shiny slows down during high winds although regular Bramblin only speed up in winds, not fly away. Then again I haven't hunted everything in this game and there might be specific mechanics I missed, so don't completely trust my intellect).
While there are Pokemon like Sudowoodo and Chansey who can take their morning runs too seriously and sprint so hard their atoms vaporize into literal nothingness, they spawn normally and only run away if you get too close (unless they are shiny in which they will jog away and not vaporize). This is different from Pokemon like Cyclizar and the Doduo line who start running around the moment they spawn from the womb and don't have different behaviors when shiny. However, they do seem to have a cycle of running for 5 seconds then pausing for 5 seconds if that's any reassuring.
Source: I paused my Gimmi-two hunt just to hunt these two (and Doduo). I actually got marks on both of them which is neat!
