First, friction between the plane and the shooting board is always an enemy. Having to overcome any unnecessary friction can cause you to push the plane out of the desired alignment. To overcome this I always apply paste wax to all of the shooting board rub surfaces (including the fence and base face) and apply paraffin to the plane. Next, because most shooting is on end grain, the plane needs to be as sharp as you can get it, set for a super fine cut, and take repeated super thin shavings. I normally start the strokes with the stock away from the plane and gently slide the stock up to contact the plane sole as the strokes continue. try to keep the plane oriented so that advancement of the stock is against the plane sole but do not push the plane away from the shooting board surfaces. If the end is out of square with the edge against the fence (horizontal?), and you are satisfied that the plane is against the plane rub surface during the full plane stroke, then the fence needs to be adjusted accordingly.
As James indicates, the condition where the end ends up being out of square with the reference stock face (if you are satisfied that the plane side is held flat against the rub surface during the full stroke) indicates that the path of the blade edge is out of square with how the stock is held (vertical?). A lot of people believe that this indicates that the reference side of the plane is out of square with the plane sole, which is probably not the case with better planes such as yours, but even planes with minor out-of-square sides can be used for shooting if the blade is adjusted as needed. Plane blades used for shooting are normally sharpened with no camber and heavy weight planes provide inertia which generally helps the process.
I have found that after going through the setup process it is best to dedicate a plane (if possible) to shooting and do check the first attempt on each work session just to be sure of the settings. Also make sure the stock has been prepared well beforehand so that the reference surfaces register well during the work. It sounds complicated but after you have done a it a few times it all becomes second nature.
One thing to consider is that traditional methods of work often sought to hide end grain or often avoided squaring ends at final dimensions just for layout operations. There are exceptions (possibly through tenons, breadboard end boards, layout of dovetails from board ends, etc.) but many joint layout operations were conducted from face and edge reference surfaces only using squares instead of using the end as a reference surface. A good example would be blind tenon layout. Why go through the process of shooting an end if the end gets sawed off later to ensure it doesn’t bottom out and to provide glue space? The idea traditionally, of course, was to know which dimensions and conditions were critical to the finished product and processes necessary to get there in in the most expeditious way. We hobbyists enjoy the luxury of doing things any way we darn well please. Have fun.