I know you need to aim left for irons and right for woods to hit it "straight." 2 questions:
1. Are you aiming your whole body left/right or just feet?
2. Are you swinging along your feet line or the target line?
All that matters is that your path is left. You can aim directly at the flag and swing left, or aim slightly left and swing along your body/feet line, whatever you like, as long as you're swinging slightly left of the targetline. (for irons)
Yes, you can swing too left. But, you might be surprised how far left you can swing and still hit it pretty straight IMO.
As to your second question, I guess maybe you could. Many people on this forum have said thats what they do (aim body left but keep face at the target). I always just aim the face straight at the target and align my body parallel to that, and just swing left enough to compensate.