My old Schon shafts were about 12.2 - 12.3mm. I switched to Z2 (11.75) and used that for a few years. Now using 314-2 or OB1, 12.75 mm. I probably ideally prefer 12.5 or so, may get my larger ones taken down a bit.
I see no difference in spin application, accuracy, etc. Only a small portion of the tip hits the CB, evidenced by chalk marks. Using 1 tip of english is still 1 tip with either shaft. I think visually the smaller shafts have that "toothpick" look for people that aren't used to them and can give the perception that you have to be more accurate. Since the thinner LD shafts have less deflection, a small amount off center shouldn't make much difference anyway.
If it helps, I switched to a slightly larger shaft just to get a more solid feeling hit. The Z2 type shafts just felt like they vibrated a bit more. I did like the conical taper though, have tried a conical or parabolic taper on a 12.3 or 12.4mm shaft, and it widens out too quickly and causes your bridge to widen uncomfortably.
BTW, I have 4 older Schon shafts I'm getting ready to sell. They are from the late 80's / early 90's and in various conditions, all around 12.1 - 12.3mm. If you wanted to take them down, you could shave a little off and smooth out whatever small dings etc. might be present in them. I will be taking pics and putting them up soon, but if you are interested PM me. That way you could keep your shaft as is, get a skinnier one, and decide from there.
Scott