Warble is definitely not a mouthpiece position issue, but is likely a voicing issue possibly exasperated by a leak. As Dr. G said amount of visible cork is arbitrary, and that is true because there is no 'industry standard' cork size that is fitted to the neck, so 1/16th" visible on one neck likely does not equal 1/16th on another neck.
For helping nail down the correct voicing you'll want to do some venting exercises. As an example play a middle C, then finger a low C but maintain the same exact note by playing the first overtone of that low C. If the normal fingering (LH 2) is sharper, then you may be pushed in too far, if flat then you may be pulled out too far. The reason I say 'may be' is because the voicing of your air (shape defined by throat position) will influence this, so it's possible to overcome the natural tuning if the voicing is off too far.
As you make mouthpiece adjustments based on that overtone exercise and then do some scales to the upper range of the horn you'll get a better idea as to where you are. Since soprano sax tends to be the highest pitched sax in the family most people play (unless you are coming from Clarinet, or maybe Oboe) the most common thing is that the voicing of the air is shaped to fit a lower pitched instrument.
If you try these things and aren't sure of what you need to do I would suggest providing a recording so we can hear, I think many of the experienced players here could better advise you on how to correct what might be happening from that rather than a verbal description of what is happening.
I am also a Series III Soprano owner, and at least in my experience (I use the curved neck fyi) this is a well balanced, and in tune horn when aligned properly and fed with an accurate voicing.
Specific to leaks, common places are the bis pad under the LH 2 finger, or the smaller Bb pad just below it not fully closing. Also G# is a good thing to check when on a very low note - low C#, B, and Bb actuate the G# mechanism so if the adjustment screw that holds the G# down is not secure that is a common leak point.