Would it be better to straighten my teeth or to fill my lips in order to even out my smile? (photos)

To the average person, these subtle differences are minimal but to you, I know they are significant.  The first question i need to ask is you have had Botox or Dermal Fillers in the past?  it seems as if the neutral zone (or lateral 1/3 aspect of your upper lip) is not symmetrical.  The upper arch also appears to be narrow.  We need to find out why this is so.  My first suggestion is to have an orthodontic and oral maxillofacial evaluation.  We need to find out whether this is a descrepancy of soft tissue, hard tissue or a combination of both.  

The shape of the soft tissues depends on the shape of the hard tissues. there is no support for your lips around the corners of your mouth, resulting in drooping of lips over there. You should have consulted a maxillofacial surgeon before your orthodontic treatment. Past is past , so the left over option is to get a fat filler into the depressed areas. Consult a good oral and maxillofacial surgeon for further advice.

You have a very narrow arch of teeth in your mouth. You might have had your bicuspid teeth extracted years ago for orthodontics, resulting in the narrowing of your arch. The teeth serve as a support of the lips, therefore, your lips have also "turned inwards", as you put it. I would suggest you have a consultation with an orthodontist to discuss your concerns.