Oh and before you mess with the wheel, try installing it backwards and see if that makes it closer to the non-drive side. That would suggest that it's really the wheel that needs fixing. If the wheel still rubs in the same way as before, you're probably looking at a dropout or frame alignment issue instead, as filtersweep suggests.

You say that you tried pushing the dropouts all the way forward. It's probably best to ignore the relative positioning of the left and right dropouts. Instead, see if you can adjust them such that the wheel becomes straight. It doesn't take much of an asymmetry at the dropout to produce a noticeable alignment issue at the rim.