put the correct design on an applique' patch of the same shirt material and add it over the mistake, (buy an extra shirt or two to get the same material) cheap out and looks pretty cool if done neatly. Can use a matching color sewdown or a contrasting color . Sewing over any other design directly is not the best solution IMHO.
On another note to remove errant stitches take killer time and you will screw some up guaranteed.
We found the best method to renove stitches was a real scalpel with quality blades and change the blade often.
try these guys really good stuff ,
http://www.tedpella.com/dissect_html/scalpel.htmbuy the non sterril blades, makes X-acto blades look like a hack saw
mooseman