your spliting just your top layer.
open up the layers tab and bounce to clips all your layers,this way you split all.
whats happening is when you split the clip and drag it or move it..the take underneath is still there.
it would be a perfect world if all your tracks were one shots and not edited(punching in,exc.)
hope this helps.
i may have misunderstood your post.
try changing your snap settings.sounds like that may be your problem