Originally Posted by
RubeRad
I sure hope it doesn't come down to filing for me! I don't think this is it, because as far as I can see with eyeball, the wheels are well centered and in the same plane as the frame
DO NOT file the dropouts just because that worked for someone else. Besides, that's an illogical solution for you, considering that the bike previously rode no-hands.
You should first check dish, and certainly nothing wrong with checking headset adjustment, though I don't know what about it would cause deflection in only one direction.
If your eyes are good it is not that difficult to mostly determine where the problem lies with no tools at all. Either use a stand that allows you to rotate the bike to an inverted position or get a friend to help you. As you say the wheels are already centered in the fork and rear triangle, first sight along the wheels from front to back, turning the front wheel as needed to align it. First look across the top of both wheels to see if one is canted off vertical in relationship to the other. Then site along the sides of the front wheel toward the rear, observing to what extent you can see the back end of the rear wheel. Then do the same along both sides of the front wheel while looking toward the down and top tubes. Add the string test and you should be able to determine whether the problem is front or rear.